Quiver varieties and finite dimensional representations of quantum affine algebras

Abstract

We study finite dimensional representations of the quantum affine algebra ${\mathbf{U}}_q(\widehat{\mathfrak{g}})$ using geometry of quiver varieties introduced by the author. As an application, we obtain character formulas expressed in terms of intersection cohomologies of quiver varieties.

Introduction

Let ${\mathfrak{g}}$ be a simple finite dimensional Lie algebra of type $ADE$, let $\widehat{{\mathfrak{g}}}$ be the corresponding (untwisted) affine Lie algebra, and let ${\mathbf{U}}_q(\widehat{\mathfrak{g}})$ be its quantum enveloping algebra of Drinfel’d-Jimbo, or the quantum affine algebra for short. In this paper we study finite dimensional representations of ${\mathbf{U}}_q(\widehat{\mathfrak{g}})$, using geometry of quiver varieties which were introduced in 294445.

There is a large amount of literature on finite dimensional representations of ${\mathbf{U}}_q(\widehat{\mathfrak{g}})$; see for example 110182528 and the references therein. A basic result relevant to us is due to Chari-Pressley 11: irreducible finite dimensional representations are classified by an $n$-tuple of polynomials, where $n$ is the rank of ${\mathfrak{g}}$. This result was announced for Yangian earlier by Drinfel’d 15. Hence the polynomials are called Drinfel’d polynomials. However, not much is known about the properties of irreducible finite dimensional representations, say their dimensions, tensor product decomposition, etc.

Quiver varieties are generalizations of moduli spaces of instantons (anti-self-dual connections) on certain classes of real $4$-dimensional hyper-Kähler manifolds, called ALE spaces 29. They can be defined for any finite graph, but we are concerned for the moment with the Dynkin graph of type $ADE$ corresponding to ${\mathfrak{g}}$. Motivated by results of Ringel 47 and Lusztig 33, the author has been studying their properties 4445. In particular, it was shown that there is a homomorphism

$$\begin{equation*} {\mathbf{U}}({\mathfrak{g}}) \to H_{\operatorname {top}}(Z({\mathbf{w}}),{\mathbb{C}}), \end{equation*}$$

where ${\mathbf{U}}({\mathfrak{g}})$ is the universal enveloping algebra of ${\mathfrak{g}}$, $Z({\mathbf{w}})$ is a certain lagrangian subvariety of the product of quiver varieties (the quiver variety depends on a choice of a dominant weight ${\mathbf{w}}$), and $H_{\operatorname {top}}(\ ,{\mathbb{C}})$ denotes the top degree homology group with complex coefficients. The multiplication on the right hand side is defined by the convolution product.

During the study, it became clear that the quiver varieties are analogous to the cotangent bundle $T^*\mathcal{B}$ of the flag variety $\mathcal{B}$. The lagrangian subvariety $Z({\mathbf{w}})$ is an analogue of the Steinberg variety $Z = T^*\mathcal{B}\times _{\mathcal{N}} T^*\mathcal{B}$, where $\mathcal{N}$ is the nilpotent cone and $T^*\mathcal{B}\to \mathcal{N}$ is the Springer resolution. The above mentioned result is an analogue of Ginzburg’s lagrangian construction of the Weyl group $W$ 20. If we replace homology group by equivariant $K$-homology group in the case of $T^*\mathcal{B}$, we get the affine Hecke algebra $H_q$ instead of $W$ as was shown by Kazhdan-Lusztig 26 and Ginzburg 13. Thus it became natural to conjecture that an equivariant $K$-homology group of the quiver variety gave us the quantum affine algebra ${\mathbf{U}}_q(\widehat{\mathfrak{g}})$. After the author wrote 44, many people suggested this conjecture to him, for example Kashiwara, Ginzburg, Lusztig and Vasserot.

A geometric approach to finite dimensional representations of ${\mathbf{U}}_q(\widehat{\mathfrak{g}})$ (when ${\mathfrak{g}} = {\mathfrak{sl}}_n$) was given by Ginzburg-Vasserot 2158. They used the cotangent bundle of the $n$-step partial flag variety, which is an example of a quiver variety of type $A$. Thus their result can be considered as a partial solution to the conjecture.

In 23 Grojnowski constructed the lower-half part ${\mathbf{U}}_q(\widehat{\mathfrak{g}})^-$ of ${\mathbf{U}}_q(\widehat{\mathfrak{g}})$ on equivariant $K$-homology of a certain lagrangian subvariety of the cotangent bundle of a variety ${\mathbf{E}_{\mathbf{d}}}$. This ${\mathbf{E}_{\mathbf{d}}}$ was used earlier by Lusztig for the construction of canonical bases on the lower-half part ${\mathbf{U}}_q({\mathfrak{g})}^-$ of the quantized enveloping algebra ${\mathbf{U}}_q({\mathfrak{g})}$. Grojnowski’s construction was motivated in part by Tanisaki’s result 52: a homomorphism from the finite Hecke algebra to the equivariant $K$-homology of the Steinberg variety is defined by assigning to perverse sheaves (or more precisely Hodge modules) on $\mathcal{B}$ their characteristic cycles. In the same way, he considered characteristic cycles of perverse sheaves on ${\mathbf{E}_{\mathbf{d}}}$. Thus he obtained a homomorphism from ${\mathbf{U}}_q({\mathfrak{g})}^-$ to $K$-homology of the lagrangian subvariety. This lagrangian subvariety contains a lagrangian subvariety of the quiver variety as an open subvariety. Thus his construction was a solution to ‘half’ of the conjecture.

Later Grojnowski wrote an ‘advertisement’ of his book on the full conjecture 24. Unfortunately, details were not explained, and his book is not published yet.

The purpose of this paper is to solve the conjecture affirmatively, and to derive results whose analogues are known for $H_q$. Recall that Kazhdan-Lusztig 26 gave a classification of simple modules of $H_q$, using the above mentioned $K$-theoretic construction. Our analogue is the Drinfel’d-Chari-Pressley classification. Also Ginzburg gave a character formula, called a $p$-adic analogue of the Kazhdan-Lusztig multiplicity formula 13. (See the introduction in 13 for a more detailed account and historical comments.) We prove a similar formula for ${\mathbf{U}}_q(\widehat{\mathfrak{g}})$ in this paper.

Let us describe the contents of this paper in more detail. In §1 we recall a new realization of ${\mathbf{U}}_q(\widehat{\mathfrak{g}})$, called Drinfel’d realization 15. It is more suitable than the original one for our purpose, or rather, we can consider it as a definition of ${\mathbf{U}}_q(\widehat{\mathfrak{g}})$. We also introduce the quantum loop algebra ${\mathbf{U}}_q({\mathbf{L}}{\mathfrak{g}})$, which is a subquotient of ${\mathbf{U}}_q(\widehat{\mathfrak{g}})$, i.e., the quantum affine algebra without central extension and the degree operator. Since the central extension acts trivially on finite dimensional representations, we study ${\mathbf{U}}_q({\mathbf{L}}{\mathfrak{g}})$ rather than ${\mathbf{U}}_q(\widehat{\mathfrak{g}})$. Introducing a certain ${\mathbb{Z}}[q,q^{-1}]$-subalgebra ${\mathbf{U}}^{{\mathbb{Z}}}_q({\mathbf{L}}{\mathfrak{g}})$ of ${\mathbf{U}}_q({\mathbf{L}}{\mathfrak{g}})$, we define a specialization ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})$ of ${\mathbf{U}}_q({\mathbf{L}}{\mathfrak{g}})$ at $q = \varepsilon$. This ${\mathbf{U}}^{{\mathbb{Z}}}_q({\mathbf{L}}{\mathfrak{g}})$ was originally introduced by Chari-Pressley 12 for the study of finite dimensional representations of ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})$ when $\varepsilon$ is a root of unity. Then we recall basic results on finite dimensional representations of ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})$. We introduce several concepts, such as l-weights, l-dominant, l-highest weight modules, l-fundamental representation, etc. These are analogues of the same concepts without l for ${\mathbf{U}}_{\varepsilon }({\mathfrak{g}})$-modules. ‘l’ stands for the loop. In the literature, some of these concepts were used without ‘l’.

In §2 we introduce two types of quiver varieties, ${\mathfrak{M}}({\mathbf{w}})$, ${\mathfrak{M}}_0(\infty ,{\mathbf{w}})$ (both depend on a choice of a dominant weight ${\mathbf{w}}= \sum w_k \Lambda _k$). They are analogues of $T^*\mathcal{B}$ and the nilpotent cone $\mathcal{N}$ respectively, and have the following properties:

(1)

${\mathfrak{M}}({\mathbf{w}})$ is a nonsingular quasi-projective variety, having many components of various dimensions.

(2)

${\mathfrak{M}}_0(\infty ,{\mathbf{w}})$ is an affine algebraic variety, not necessarily irreducible.

(3)

Both ${\mathfrak{M}}({\mathbf{w}})$ and ${\mathfrak{M}}_0(\infty ,{\mathbf{w}})$ have a $G_{\mathbf{w}}\times {\mathbb{C}}^*$-action, where $G_{\mathbf{w}}= \prod \operatorname {GL}_{w_k}({\mathbb{C}})$.

(4)

There is a $G_{\mathbf{w}}\times {\mathbb{C}}^*$-equivariant projective morphism $\pi \colon {\mathfrak{M}}({\mathbf{w}})\to {\mathfrak{M}}_0(\infty ,{\mathbf{w}})$.

In §3–§8 we prepare some results on quiver varieties and $K$-theory which we use in later sections.

In §9–§11 we consider an analogue of the Steinberg variety $Z({\mathbf{w}})= {\mathfrak{M}}({\mathbf{w}})\times _{{\mathfrak{M}}_0(\infty ,{\mathbf{w}})}{\mathfrak{M}}({\mathbf{w}})$ and its equivariant $K$-homology $K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(Z({\mathbf{w}}))$. We construct an algebra homomorphism

$$\begin{equation*} {\mathbf{U}}_q({\mathbf{L}}{\mathfrak{g}})\to K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(Z({\mathbf{w}}))\otimes _{{\mathbb{Z}}[q,q^{-1}]}{\mathbb{Q}}(q). \end{equation*}$$

We first define images of generators in §9, and check the defining relations in §10 and §11. Unlike the case of the affine Hecke algebra, where $H_q$ is isomorphic to $K^{G\times {\mathbb{C}}^*}(Z)$ ($Z =$ the Steinberg variety), this homomorphism is not an isomorphism, neither injective nor surjective.

In §12 we show that the above homomorphism induces a homomorphism

$$\begin{equation*} {\mathbf{U}}^{{\mathbb{Z}}}_q({\mathbf{L}}{\mathfrak{g}})\to K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(Z({\mathbf{w}}))/\operatorname {torsion}. \end{equation*}$$

(It is natural to expect that ${\mathbf{U}}^{{\mathbb{Z}}}_q({\mathbf{L}}{\mathfrak{g}})$ is an integral form of ${\mathbf{U}}_q({\mathbf{L}}{\mathfrak{g}})$ and that $K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(Z({\mathbf{w}}))$ is torsion-free, but we do not have the proofs.)

In §13 we introduce a standard module $M_{x,a}$. It depends on the choice of a point $x\in {\mathfrak{M}}_0(\infty ,{\mathbf{w}})$ and a semisimple element $a = (s,\varepsilon )\in G_{\mathbf{w}}\times {\mathbb{C}}^*$ such that $x$ is fixed by $a$. The parameter $\varepsilon$ corresponds to the specialization $q = \varepsilon$, while $s$ corresponds to Drinfel’d polynomials. In this paper, we assume $\varepsilon$ is not a root of unity, although most of our results hold even in that case (see Remark 14.3.9). Let $A$ be the Zariski closure of $a^{{\mathbb{Z}}}$. We define $M_{x,a}$ as the specialized equivariant $K$-homology $K^A({\mathfrak{M}}({\mathbf{w}})_x)\otimes _{R(A)}{\mathbb{C}}_a$, where ${\mathfrak{M}}({\mathbf{w}})_x$ is a fiber of ${\mathfrak{M}}({\mathbf{w}})\to {\mathfrak{M}}_0(\infty ,{\mathbf{w}})$ at $x$, and ${\mathbb{C}}_a$ is an $R(A)$-algebra structure on ${\mathbb{C}}$ determined by $a$. By the convolution product, $M_{x,a}$ has a $K^A(Z({\mathbf{w}}))\otimes _{R(A)}{\mathbb{C}}_a$-module structure. Thus it has a ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})$-module structure by the above homomorphism. By the localization theorem of equivariant $K$-homology due to Thomason 55, $M_{x,a}$ is isomorphic to the complexified (non-equivariant) $K$-homology $K({\mathfrak{M}}({\mathbf{w}})_x^A)\otimes {\mathbb{C}}$ of the fixed point set ${\mathfrak{M}}({\mathbf{w}})_x^A$. Moreover, it is isomorphic to $H_*({\mathfrak{M}}({\mathbf{w}})_x^A,{\mathbb{C}})$ via the Chern character homomorphism thanks to a result in §7. We also show that $M_{x,a}$ is a finite dimensional l-highest weight module. As a usual argument for Verma modules, $M_{x,a}$ has the unique (nonzero) simple quotient. The author conjectures that $M_{x,a}$ is a tensor product of l-fundamental representations in some order. This is proved when the parameter is generic in §14.1.

In §14 we show that the standard modules $M_{x,a}$ and $M_{y,a}$ are isomorphic if and only if $x$ and $y$ are contained in the same stratum. Here the fixed point set ${\mathfrak{M}}_0(\infty ,{\mathbf{w}})^A$ has a stratification ${\mathfrak{M}}_0(\infty ,{\mathbf{w}})^A = \bigsqcup _\rho {\mathfrak{M}_0^{\operatorname {reg}}}(\rho )$ defined in §4. Furthermore, we show that the index set $\{ \rho \}$ of the stratum coincides with the set $\mathcal{P} = \{ P \}$ of l-dominant l-weights of $M_{0,a}$, the standard module corresponding to the central fiber $\pi ^{-1}(0)$. Let us denote by $\rho _P$ the index corresponding to $P$. Thus we may denote $M_{x,a}$ and its unique simple quotient by $M(P)$ and $L(P)$ respectively if $x$ is contained in the stratum ${\mathfrak{M}_0^{\operatorname {reg}}}(\rho _P)$ corresponding to an l-dominant l-weight $P$. We prove the multiplicity formula

$$\begin{equation*} \left[ M(P) : L(Q) \right] = \dim H^*(i_x^! IC({\mathfrak{M}_0^{\operatorname {reg}}}(\rho _Q))), \end{equation*}$$

where $x$ is a point in ${\mathfrak{M}_0^{\operatorname {reg}}}(\rho _P)$, $i_x\colon \{x\}\to {\mathfrak{M}}_0(\infty ,{\mathbf{w}})^A$ is the inclusion, and $IC({\mathfrak{M}_0^{\operatorname {reg}}}(\rho _Q))$ is the intersection cohomology complex attached to ${\mathfrak{M}_0^{\operatorname {reg}}}(\rho _Q)$ and the constant local system ${\mathbb{C}}_{{\mathfrak{M}_0^{\operatorname {reg}}}(\rho _Q)}$.

Our result is simpler than the case of the affine Hecke algebra: nonconstant local systems never appear. This phenomenon corresponds to an algebraic result that all modules are l-highest weight. It compensates for the difference of ${\mathbf{U}}^{{\mathbb{Z}}}_q({\mathbf{L}}{\mathfrak{g}})$ and $K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(Z({\mathbf{w}}))$ during the proof of the multiplicity formula.

If ${\mathfrak{g}}$ is of type $A$, then ${\mathfrak{M}}_0(\infty ,{\mathbf{w}})^A$ coincides with a product of varieties ${\mathbf{E}_{\mathbf{d}}}$ studied by Lusztig 33, where the underlying graph is of type $A$. In particular, the Poincaré polynomial of $H^*(i_x^! IC({\mathfrak{M}_0^{\operatorname {reg}}}(\rho _Q)))$ is a Kazhdan-Lusztig polynomial for a Weyl group of type $A$. We should have a combinatorial algorithm to compute Poincaré polynomials of $H^*(i_x^! IC({\mathfrak{M}_0^{\operatorname {reg}}}(\rho _Q)))$ for general $\mathfrak{g}$.

Once we know $\dim H^*(i_x^! IC({\mathfrak{M}_0^{\operatorname {reg}}}(\rho _Q)))$, information about $L(P)$ can be deduced from information about $M(P)$, which is easier to study. For example, consider the following problems:

(1)

Compute Frenkel-Reshetikhin’s $q$-characters 18.

(2)

Decompose restrictions of finite dimensional ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})$-modules to ${\mathbf{U}}_{\varepsilon }({\mathfrak{g}})$-modules (see 28).

These problems for $M(P)$ are easier than those for $L(P)$, and we have the following answers.

Frenkel-Reshetikhin’s $q$-characters are generating functions of dimensions of l-weight spaces (see §13.5). In §13.5 we show that these dimensions are Euler numbers of connected components of ${\mathfrak{M}}({\mathbf{w}})^A$ for standard modules $M_{0,a}$. As an application, we prove a conjecture in 18 for $\mathfrak{g}$ of type $ADE$ (Proposition 13.5.2). These Euler numbers should be computable.

Let $\operatorname {Res}M(P)$ be the restriction of $M(P)$ to a ${\mathbf{U}}_{\varepsilon }({\mathfrak{g}})$-module. In §15 we show the multiplicity formula

$$\begin{equation*} \left[ \operatorname {Res}M(P) : L({\mathbf{w}}-{\mathbf{v}}) \right] = \dim H^*(i_x^! IC({\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}}))), \end{equation*}$$

where ${\mathbf{v}}$ is a weight such that ${\mathbf{w}}- {\mathbf{v}}$ is dominant, $L({\mathbf{w}}-{\mathbf{v}})$ is the corresponding irreducible finite dimensional module (these are concepts for usual ${\mathfrak{g}}$ without ‘l’), $x$ is a point in ${\mathfrak{M}_0^{\operatorname {reg}}}(\rho _P)$, $i_x\colon \{x\}\to {\mathfrak{M}}_0(\infty ,{\mathbf{w}})$ is the inclusion, ${\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}})$ is a stratum of ${\mathfrak{M}}(\infty ,{\mathbf{w}})$, and $IC({\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}}))$ is the intersection cohomology complex attached to ${\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}})$ and the constant local system ${\mathbb{C}}_{{\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}})}$.

If ${\mathfrak{g}}$ is of type $A$, then the stratum ${\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}})$ coincides with a nilpotent orbit cut out by Slodowy’s transversal slice 44, 8.4. The Poincaré polynomials of $H^*(i_x^! IC({\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}})))$ were calculated by Lusztig 30 and coincide with Kostka polynomials. This result is compatible with the conjecture that $M(P)$ is a tensor product of l-fundamental representations, for the restriction of an l-fundamental representation is simple for type $A$, and Kostka polynomials give tensor product decompositions. We should have a combinatorial algorithm to compute Poincaré polynomials of $H^*(i_x^!IC({\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}})))$ for general $\mathfrak{g}$.

We give two examples where ${\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}})$ can be described explicitly.

Consider the case that ${\mathbf{w}}$ is a fundamental weight of type $A$, or more generally a fundamental weight such that the label of the corresponding vertex of the Dynkin diagram is $1$. Then it is easy to see that the corresponding quiver variety ${\mathfrak{M}}_0(\infty ,{\mathbf{w}})$ consists of a single point $0$. Thus $\operatorname {Res}M(P)$ remains irreducible in this case.

If ${\mathbf{w}}$ is the highest weight of the adjoint representation, the corresponding ${\mathfrak{M}}_0(\infty ,{\mathbf{w}})$ is a simple singularity ${\mathbb{C}}^2/\Gamma$, where $\Gamma$ is a finite subgroup of $\operatorname {SL}_2({\mathbb{C}})$ of the type corresponding to ${\mathfrak{g}}$. Then ${\mathfrak{M}}_0(\infty ,{\mathbf{w}})$ has two strata $\{0\}$ and $({\mathbb{C}}^2\setminus \{0\})/\Gamma$. The intersection cohomology complexes are constant sheaves. Hence we have

$$\begin{equation*} \operatorname {Res}M(P) = L({\mathbf{w}})\oplus L(0). \end{equation*}$$

These two results were shown by Chari-Pressley 9 by a totally different method.

As we mentioned, the quantum affine algebra ${\mathbf{U}}_q(\widehat{\mathfrak{g}})$ has another realization, called the Drinfel’d new realization. This Drinfel’d construction can be applied to any symmetrizable Kac-Moody algebra ${\mathfrak{g}}$, not necessarily a finite dimensional one. This generalization also fits our result, since quiver varieties can be defined for arbitrary finite graphs. If we replace finite dimensional representations by l-integrable representations, parts of our result can be generalized to a Kac-Moody algebra ${\mathfrak{g}}$, at least when it is symmetric. For example, we generalize the Drinfel’d-Chari-Pressley parametrization. A generalization of the multiplicity formula requires further study.

If ${\mathfrak{g}}$ is an affine Lie algebra, then ${\mathbf{U}}_q(\widehat{\mathfrak{g}})$ is the quantum affinization of the affine Lie algebra. It is called a double loop algebra, or toroidal algebra, and has been studied by various people; see for example 22484956 and the references therein. A first step to a geometric approach to the toroidal algebra using quiver varieties for the affine Dynkin graph of type $\widetilde{A}$ was given by M. Varagnolo and E. Vasserot 57. In fact, quiver varieties for affine Dynkin graphs are moduli spaces of instantons (or torsion free sheave) on ALE spaces. Thus these cases are relevant to the original motivation, i.e., a study of the relation between $4$-dimensional gauge theory and representation theory. In some cases, these quiver varieties coincide with Hilbert schemes of points on ALE spaces, for which many results have been obtained (see 46). We will return to this in the future.

If we replace equivariant $K$-homology by equivariant homology, we should get the Yangian $Y({\mathfrak{g}})$ instead of ${\mathbf{U}}_q(\widehat{\mathfrak{g}})$. This conjecture is motivated again by the analogy of quiver varieties with $T^*\mathcal{B}$. The equivariant homology of $T^*\mathcal{B}$ gives the graded Hecke algebra 32, which is an analogue of $Y({\mathfrak{g}})$ for $H_q$. As an application, the affirmative solution of the conjecture implies that the representation theory of ${\mathbf{U}}_q(\widehat{\mathfrak{g}})$ and that of the Yangian are the same. This has been believed by many people, but there is no written proof.

While the author was preparing this paper, he was informed that Frenkel-Mukhin 17 proved the conjecture in 18 (Proposition 13.5.2) for general ${\mathfrak{g}}$.

Acknowledgement.

Part of this work was done while the author enjoyed the hospitality of the Institute for Advanced Study. The author is grateful to G. Lusztig for his interest and encouragement.

1. Quantum affine algebra

In this section, we give a quick review for the definitions of the quantized universal enveloping algebra ${\mathbf{U}}_q({\mathfrak{g})}$ of the Kac-Moody algebra ${\mathfrak{g}}$ associated with a symmetrizable generalized Cartan matrix, its affinization ${\mathbf{U}}_q(\widehat{\mathfrak{g}})$, and the associated loop algebra ${\mathbf{U}}_q({\mathbf{L}}{\mathfrak{g}})$. Although the algebras defined via quiver varieties are automatically symmetric, we treat the nonsymmetric case also for completeness.

1.1. Quantized universal enveloping algebra

Let $q$ be an indeterminate. For nonnegative integers $n\ge r$, define

$$\begin{equation} \begin{split} [n]_q \overset{\operatorname {\scriptstyle def.}}{=}\frac{q^n - q^{-n}}{q - q^{-1}}, \quad [n]_q !& \overset{\operatorname {\scriptstyle def.}}{=}\begin{cases}[n] _q [n-1]_q \cdots [2]_q [1]_q &(n > 0),\\ 1 &(n=0), \end{cases}\\ \begin{bmatrix} n \\ r \end{bmatrix}_q &\overset{\operatorname {\scriptstyle def.}}{=}\frac{[n]_q !}{[r]_q! [n-r]_q!}. \end{split}{\tag{1.1.1}} \end{equation}$$

Suppose that the following data are given:

(1)

$P$ : free ${\mathbb{Z}}$-module (weight lattice),

(2)

$P^* = \operatorname {Hom}_{{\mathbb{Z}}}(P, {\mathbb{Z}})$ with a natural pairing $\langle \ , \ \rangle \colon P\otimes P^*\to {\mathbb{Z}}$,

(3)

an index set $I$ of simple roots

(4)

$\alpha _k\in P$ ($k \in I$) (simple root),

(5)

$h_k \in P^*$ ($k \in I$) (simple coroot),

(6)

a symmetric bilinear form $(\ ,\ )$ on $P$.

These are required to satisfy the following:

(a)

$\langle h_k, \lambda \rangle = 2(\alpha _k, \lambda )/(\alpha _k,\alpha _k)$ for $k \in I$ and $\lambda \in P$,

(b)

${\mathbf{C}}\overset{\operatorname {\scriptstyle def.}}{=}(\langle h_k, \alpha _l \rangle )_{k,l}$ is a symmetrizable generalized Cartan matrix, i.e., $\langle h_k, \alpha _k\rangle = 2$, and $\langle h_k, \alpha _l\rangle \in {\mathbb{Z}}_{\le 0}$ and $\langle h_k, \alpha _l\rangle = 0 \Longleftrightarrow \langle h_l, \alpha _k\rangle = 0$ for $k\ne l$,

(c)

$(\alpha _k,\alpha _k)\in 2{\mathbb{Z}}_{> 0}$,

(d)

$\{\alpha _k\}_{k\in I}$ are linearly independent,

(e)

there exists $\Lambda _k\in P$ ($k \in I$) such that $\langle h_l, \Lambda _k\rangle = \delta _{kl}$ (fundamental weight).

The quantized universal enveloping algebra ${\mathbf{U}}_q({\mathfrak{g})}$ of the Kac-Moody algebra is the ${\mathbb{Q}}(q)$-algebra generated by $e_k$, $f_k$ ($k \in I$), $q^h$ ($h\in P^*$) with relations

$$\begin{gather} q^0 = 1, \quad q^h q^{h'} = q^{h+h'},\tag{1.1.2}\\ q^h e_k q^{-h} = q^{\langle h, \alpha _k\rangle } e_k,\quad q^h f_k q^{-h} = q^{-\langle h, \alpha _k\rangle } f_k,\tag{1.1.3}\\ e_k f_l - f_l e_k = \delta _{kl} \frac{q^{(\alpha _k,\alpha _k)h_k/2}-q^{-(\alpha _k,\alpha _k)h_k/2}}{q_k - q_k^{-1}},\tag{1.1.4}\\ \sum _{p=0}^{b}(-1)^p \begin{bmatrix} b \\ p\end{bmatrix}_{q_k} e_k^p e_l e_k^{b-p} = \sum _{p=0}^{b}(-1)^p \begin{bmatrix} b \\ p\end{bmatrix}_{q_k} f_k^p f_l f_k^{b-p} = 0 \quad \text{for $k\ne l$,} \tag{1.1.5} \end{gather}$$

where $q_k = q^{(\alpha _k,\alpha _k)/2}$, $b = 1 - \langle h_k,\alpha _l\rangle$.

Let ${\mathbf{U}}_q({\mathfrak{g})}^+$ (resp. ${\mathbf{U}}_q({\mathfrak{g})}^-$) be the ${\mathbb{Q}}(q)$-subalgebra of ${\mathbf{U}}_q({\mathfrak{g})}$ generated by the elements $e_k$ (resp. $f_k$). Let ${\mathbf{U}}_q({\mathfrak{g})}^0$ be the ${\mathbb{Q}}(q)$-subalgebra generated by elements $q^h$ ($h\in P^*$). Then we have the triangle decomposition 36, 3.2.5:

$$\begin{equation} {\mathbf{U}}_q({\mathfrak{g})}\cong {\mathbf{U}}_q({\mathfrak{g})}^+ \otimes {\mathbf{U}}_q({\mathfrak{g})}^0 \otimes {\mathbf{U}}_q({\mathfrak{g})}^- . {\tag{1.1.6}} \end{equation}$$

Let $e_k^{(n)}\overset{\operatorname {\scriptstyle def.}}{=}e_k^n/[n]_{q_k}!$ and $f_k^{(n)}\overset{\operatorname {\scriptstyle def.}}{=}f_k^n/[n]_{q_k}!$. Let ${\mathbf{U}}^{{\mathbb{Z}}}_q({\mathfrak{g}})$ be the ${\mathbb{Z}}[q,q^{-1}]$-subalgebra of ${\mathbf{U}}_q({\mathfrak{g})}$ generated by elements $e_k^{(n)}$, $f_k^{(n)}$, $q^h$ for $k\in I$, $n\in {\mathbb{Z}}_{> 0}$, $h\in P^*$. It is known that ${\mathbf{U}}^{{\mathbb{Z}}}_q({\mathfrak{g}})$ is an integral form of ${\mathbf{U}}_q({\mathfrak{g})}$, i.e., the natural map ${\mathbf{U}}^{{\mathbb{Z}}}_q({\mathfrak{g}})\otimes _{{\mathbb{Z}}[q,q^{-1}]}{\mathbb{Q}}(q) \to {\mathbf{U}}_q({\mathfrak{g})}$ is an isomorphism. (See 10, 9.3.1.) For $\varepsilon \in {\mathbb{C}}^*$, let us define ${\mathbf{U}}_{\varepsilon }({\mathfrak{g}})$ as ${\mathbf{U}}^{{\mathbb{Z}}}_q({\mathfrak{g}})\otimes _{{\mathbb{Z}}[q,q^{-1}]}{\mathbb{C}}$ via the algebra homomorphism ${\mathbb{Z}}[q,q^{-1}]\to {\mathbb{C}}$ that takes $q$ to $\varepsilon$. It will be called the specialized quantized enveloping algebra. We say a ${\mathbf{U}}_q({\mathfrak{g})}$-module $M$ (defined over ${\mathbb{Q}}(q)$) is a highest weight module with highest weight $\Lambda \in P$ if there exists a vector $m_0\in M$ such that

$$\begin{align} & e_k\ast m_0 = 0, \qquad {\mathbf{U}}_q({\mathfrak{g})}^- \ast m_0 = M, \tag{1.1.7}\\ & q^h\ast m_0 = q^{\langle h, \Lambda \rangle } m_0 \qquad \text{for any $h\in P^*$.} \tag{1.1.8} \end{align}$$

Then there exists a direct sum decomposition $M = \bigoplus _{\lambda \in P} M_\lambda$ (weight space decomposition) where $M_\lambda \overset{\operatorname {\scriptstyle def.}}{=}\{ m\mid q^h\cdot v = q^{\langle h, \lambda \rangle } m$ for any $h\in P^*$}. By using the triangular decomposition 1.1.6, one can show that the simple highest weight ${\mathbf{U}}_q({\mathfrak{g})}$-module is determined uniquely by $\Lambda$.

We say a ${\mathbf{U}}_q({\mathfrak{g})}$-module $M$ (defined over ${\mathbb{Q}}(q)$) is integrable if $M$ has a weight space decomposition $M = \bigoplus _{\lambda \in P} M_\lambda$ with $\dim M_\lambda < \infty$, and for any $m\in M$, there exists $n_0\ge 1$ such that $e_{k}^{n} \ast m = f_{k}^{n}\ast m = 0$ for all $k\in I$ and $n\ge n_0$.

The (unique) simple highest weight ${\mathbf{U}}_q({\mathfrak{g})}$-module with highest weight $\Lambda$ is integrable if and only if $\Lambda$ is a dominant integral weight $\Lambda$, i.e., $\langle \Lambda , h_k\rangle \in {\mathbb{Z}}_{\ge 0}$ for any $k\in I$ (36, 3.5.6, 3.5.8). In this case, the integrable highest weight ${\mathbf{U}}_q$-module with highest weight $\Lambda$ is denoted by ${L}(\Lambda )$.

For a ${\mathbf{U}}_{\varepsilon }({\mathfrak{g}})$-module $M$ (defined over ${\mathbb{C}}$), we define highest weight modules, integrable modules, etc. in a similar way.

Suppose $\Lambda$ is dominant. Let ${L}(\Lambda )^{\mathbb{Z}}\overset{\operatorname {\scriptstyle def.}}{=}{\mathbf{U}}^{{\mathbb{Z}}}_q({\mathfrak{g}})\ast m_0$, where $m_0$ is the highest weight vector. It is known that the natural map ${L}(\Lambda )^{\mathbb{Z}}\otimes _{{\mathbb{Z}}[q,q^{-1}]}{\mathbb{Q}}(q)\to {L}(\Lambda )$ is an isomorphism and ${L}(\Lambda )^{\mathbb{Z}}\otimes _{{\mathbb{Z}}[q,q^{-1}]}{\mathbb{C}}$ is the simple integrable highest weight module of the corresponding Kac-Moody algebra ${\mathfrak{g}}$ with highest weight $\Lambda$, where ${\mathbb{Z}}[q,q^{-1}]\to {\mathbb{C}}$ is the homomorphism that sends $q$ to $1$ (36, Chapter 14 and 33.1.3). Unless $\varepsilon$ is a root of unity, the simple integrable highest weight ${\mathbf{U}}_{\varepsilon }({\mathfrak{g}})$-module is the specialization of ${L}(\Lambda )^{\mathbb{Z}}$ (10, 10.1.14, 10.1.15).

1.2. Quantum affine algebra

The quantum affinization ${\mathbf{U}}_q(\widehat{\mathfrak{g}})$ of ${\mathbf{U}}_q({\mathfrak{g})}$ (or simply quantum affine algebra) is an associative algebra over ${\mathbb{Q}}(q)$ generated by $e_{k,r}, f_{k,r}$ ($k\in I$, $r\in {\mathbb{Z}}$), $q^h$ ($h \in P^*$), $q^{\pm c/2}$, $q^{\pm d}$, and $h_{k,m}$ ($k\in I$, $m\in {\mathbb{Z}}\setminus \{0\}$) with the following defining relations:

$$\begin{gather} \text{$q^{\pm c/2}$ is central,} {\tag{1.2.1}}\\ q^0 = 1, \quad q^h q^{h'} = q^{h+h'}, \quad [q^h, h_{k,m}] = 0, \quad q^{d} q^{-d} = 1, \quad q^{c/2} q^{-c/2} = 1, {\tag{1.2.2}}\\ \psi _k^\pm (z) \psi _l^\pm (w) = \psi _l^\pm (w) \psi _k^\pm (z), {\tag{1.2.3}}\\ \psi _k^-(z) \psi _l^+(w) = \frac{(z - q^{-(\alpha _k,\alpha _l)}q^c w) (z - q^{(\alpha _k,\alpha _l)}q^{-c} w)}{(z - q^{(\alpha _k,\alpha _l)}q^c w) (z - q^{-(\alpha _k,\alpha _l)}q^{-c} w)} \psi _l^+(w) \psi _k^-(z) ,{\tag{1.2.4}}\\ \begin{split} & [ q^{d}, q^h ] = 0, \quad q^d h_{k,m} q^{-d} = q^m h_{k,m},\\ & q^d e_{k,r} q^{-d} = q^r e_{k,r}, \quad q^d f_{k,r} q^{-d} = q^r f_{k,r}, \end{split} {\tag{1.2.5}}\\ q^h e_{k,r} q^{-h} = q^{\langle h,\alpha _k\rangle } e_{k,r}, \quad q^h f_{k,r} q^{-h} = q^{-\langle h,\alpha _k\rangle } f_{k,r}, {\tag{1.2.6}}\\ \begin{split} & ( q^{\pm sc/2} z - q^{\pm \langle h_k,\alpha _l\rangle } w) \psi _l^s(z) x_k^\pm (w)\\ &\qquad = ( q^{\pm \langle h_k,\alpha _l\rangle } q^{\pm sc/2} z - w) x^\pm _k(w) \psi _l^s(z), \quad \end{split}{\tag{1.2.7}}\\ \left[x_{k}^+(z), x_{l}^-(w)\right] = \frac{\delta _{kl}}{q_k - q_k^{-1}} \left\{\delta \left(q^c\frac{w}{z}\right)\psi ^+_k(q^{c/2}w) - \delta \left(q^c\frac{z}{w}\right)\psi ^-_k(q^{c/2}z)\right\}, {\tag{1.2.8}}\\ (z - q^{\pm 2} w) x_{k}^\pm (z) x_{k}^\pm (w) = (q^{\pm 2} z - w) x_{k}^\pm (w) x_{k}^\pm (z), {\tag{1.2.9}}\\ \begin{split} & \prod _{p=1}^{-\langle \alpha _k,h_l\rangle } (z - q^{\pm (b'-2p)} w) x_{k}^\pm (z) x_{l}^\pm (w)\\ &\qquad = \prod _{p=1}^{-\langle \alpha _k, h_l\rangle } (q^{\pm (b'-2p)} z - w) x_{l}^\pm (w) x_{k}^\pm (z), \quad \text{if $k\neq l$}, \end{split} {\tag{1.2.10}} \end{gather}$$

$$\begin{multline} \sum _{\sigma \in S_b} \sum _{p=0}^{b}(-1)^p \begin{bmatrix} b \\ p\end{bmatrix}_{q_k} x_{k}^\pm (z_{\sigma (1)})\cdots x_{k}^\pm (z_{\sigma (p)}) x_{l}^\pm (w) x_{k}^\pm (z_{\sigma (p+1)})\\ \cdots x_{k}^\pm (z_{\sigma (b)}) = 0, \quad \text{if $k\neq l$,} {\tag{1.2.11}} \end{multline}$$ where $q_k = q^{(\alpha _k,\alpha _k)/2}$, $s = \pm$, $b = 1-\langle h_k, \alpha _l\rangle$, $b' = - (\alpha _k,\alpha _l)$, and $S_b$ is the symmetric group of $b$ letters. Here $\delta (z)$, $x_k^+(z)$, $x_k^-(z)$, $\psi ^{\pm }_{k}(z)$ are generating functions defined by

$$\begin{gather*} \delta (z) \overset{\operatorname {\scriptstyle def.}}{=}\sum _{r=-\infty }^\infty z^{r}, \qquad x_k^+(z) \overset{\operatorname {\scriptstyle def.}}{=}\sum _{r=-\infty }^\infty e_{k,r} z^{-r}, \qquad x_k^-(z) \overset{\operatorname {\scriptstyle def.}}{=}\sum _{r=-\infty }^\infty f_{k,r} z^{-r}, \\ \psi ^{\pm }_k(z) \overset{\operatorname {\scriptstyle def.}}{=}q^{\pm (\alpha _k,\alpha _k)h_k/2} \exp \left(\pm (q_k-q_k^{-1})\sum _{m=1}^\infty h_{k,\pm m} z^{\mp m}\right). \end{gather*}$$

We will also need the following generating function later:

$$\begin{equation*} p_k^\pm (z) \overset{\operatorname {\scriptstyle def.}}{=}\exp \left( - \sum _{m=1}^\infty \frac{h_{k,\pm m}}{[m]_{q_k}} z^{\mp m} \right). \end{equation*}$$

We have $\psi ^{\pm }_k(z) = q^{\pm (\alpha _k,\alpha _k)h_k/2} p_k^\pm (q_k z)/p_k^\pm (q_k^{-1} z).$

Remark 1.2.12.

When ${\mathfrak{g}}$ is finite dimensional, then $\min (\langle \alpha _k, h_l\rangle , \langle \alpha _l, h_k\rangle ) = \text{$0$ or $1$}$. Then the relation 1.2.10 reduces to the one in literature. Our generalization seems natural since we will check it later, at least for symmetric ${\mathfrak{g}}$.

Let ${\mathbf{U}}_q(\widehat{\mathfrak{g}})^+$ (resp. ${\mathbf{U}}_q(\widehat{\mathfrak{g}})^-$) be the ${\mathbb{Q}}(q)$-subalgebra of ${\mathbf{U}}_q(\widehat{\mathfrak{g}})$ generated by the elements $e_{k,r}$ (resp. $f_{k,r}$). Let ${\mathbf{U}}_q(\widehat{\mathfrak{g}})^0$ be the ${\mathbb{Q}}(q)$-subalgebra generated by the elements $q^h$, $h_{k,m}$.

The quantum loop algebra ${\mathbf{U}}_q({\mathbf{L}}{\mathfrak{g}})$ is the subalgebra of ${\mathbf{U}}_q(\widehat{\mathfrak{g}})/(q^{\pm c/2} - 1)$ generated by $e_{k,r}, f_{k,r}$ ($k\in I$, $r\in {\mathbb{Z}}$), $q^h$ ($h \in P^*$), and $h_{k,m}$ ($k\in I$, $m\in {\mathbb{Z}}\setminus \{0\}$), i.e., generators other than $q^{\pm c/2}$, $q^{\pm d}$. We will be concerned only with the quantum loop algebra, and not with the quantum affine algebra in the sequel.

There is a homomorphism ${\mathbf{U}}_q({\mathfrak{g})}\to {\mathbf{U}}_q({\mathbf{L}}{\mathfrak{g}})$ defined by

$$\begin{equation*} q^h \mapsto q^h, \quad e_k \mapsto e_{k,0}, \quad f_k \mapsto f_{k,0}. \end{equation*}$$

Let $e_{k,r}^{(n)} \overset{\operatorname {\scriptstyle def.}}{=}e_{k,r}^n / [n]_{q_k}!$ and $f_{k,r}^{(n)} \overset{\operatorname {\scriptstyle def.}}{=}f_{k,r}^n / [n]_{q_k}!$. Let ${\mathbf{U}}^{{\mathbb{Z}}}_q({\mathbf{L}}{\mathfrak{g}})$ be the ${\mathbb{Z}}[q,q^{-1}]$-subalgebra generated by $e_{k,r}^{(n)}$, $f_{k,r}^{(n)}$, $q^h$ and the coefficients of $p_k^\pm (z)$ for $k\in I$, $r\in {\mathbb{Z}}$, $n\in {\mathbb{Z}}_{> 0}$, $h\in P^*$. (It should be true that ${\mathbf{U}}^{{\mathbb{Z}}}_q({\mathbf{L}}{\mathfrak{g}})$ is free over ${\mathbb{Z}}[q,q^{-1}]$ and that the natural map ${\mathbf{U}}^{{\mathbb{Z}}}_q({\mathbf{L}}{\mathfrak{g}})\otimes _{{\mathbb{Z}}[q,q^{-1}]}{\mathbb{Q}}(q) \to {\mathbf{U}}_q({\mathbf{L}}{\mathfrak{g}})$ is an isomorphism. But the author does not know how to prove this.) This subalgebra was introduced by Chari-Pressley 12. Let ${\mathbf{U}}^{{\mathbb{Z}}}_q({\mathbf{L}}{\mathfrak{g}})^+$ (resp. ${\mathbf{U}}^{{\mathbb{Z}}}_q({\mathbf{L}}{\mathfrak{g}})^-$) be the ${\mathbb{Z}}[q,q^{-1}]$-subalgebra generated by $e_{k,r}^{(n)}$ (resp. $f_{k,r}^{(n)}$) for $k\in I$, $r\in {\mathbb{Z}}$, $n\in Z_{> 0}$. We have ${\mathbf{U}}^{{\mathbb{Z}}}_q({\mathbf{L}}{\mathfrak{g}})^\pm \subset {\mathbf{U}}^{{\mathbb{Z}}}_q({\mathbf{L}}{\mathfrak{g}})$. Let ${\mathbf{U}}^{{\mathbb{Z}}}_q({\mathbf{L}}{\mathfrak{g}})^0$ be the ${\mathbb{Z}}[q,q^{-1}]$-subalgebra generated by $q^h$, the coefficients of $p_k^\pm (z)$ and

$$\begin{equation*} \begin{bmatrix} q^{h_k}; n \\ r \end{bmatrix} \overset{\operatorname {\scriptstyle def.}}{=}\prod _{s=1}^r \frac{q^{(\alpha _k,\alpha _k)h_k/2} q_k^{n-s+1} - q^{-(\alpha _k,\alpha _k)h_k/2} q_k^{-n+s-1}}{q_k^s - q_k^{-s}} \end{equation*}$$

for all $h\in P$, $k\in I$, $n\in {\mathbb{Z}}$, $r\in {\mathbb{Z}}_{> 0}$. One can easily show that ${\mathbf{U}}^{{\mathbb{Z}}}_q({\mathbf{L}}{\mathfrak{g}})^0\subset {\mathbf{U}}^{{\mathbb{Z}}}_q({\mathbf{L}}{\mathfrak{g}})$ (see, e.g., 36, 3.1.9).

For $\varepsilon \in {\mathbb{C}}^*$, let ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})$ be the specialized quantum loop algebra defined by ${\mathbf{U}}^{{\mathbb{Z}}}_q({\mathbf{L}}{\mathfrak{g}})\otimes _{{\mathbb{Z}}[q,q^{-1}]}{\mathbb{C}}$ via the algebra homomorphism ${\mathbb{Z}}[q,q^{-1}]\to {\mathbb{C}}$ that takes $q$ to $\varepsilon$. We assume $\varepsilon$ is not a root of unity in this paper. Let ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})^\pm$ and ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})^0$ be the specializations of ${\mathbf{U}}^{{\mathbb{Z}}}_q({\mathbf{L}}{\mathfrak{g}})^\pm$ and ${\mathbf{U}}^{{\mathbb{Z}}}_q({\mathbf{L}}{\mathfrak{g}})^0$ respectively. We have a weak form of the triangular decomposition

$$\begin{equation} {\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})= {\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})^-\cdot {\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})^0\cdot {\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})^+, {\tag{1.2.13}} \end{equation}$$

which follows from the definition (cf. 12, 6.1).

We say a ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})$-module $M$ is an l-highest weight module (‘l’ stands for the loop) with l-highest weight $(\Lambda , (\Psi ^{\pm }_{k}(z))_k)$ (where $\Lambda \in P$, $(\Psi ^{\pm }_{k}(z))_k\in {\mathbb{C}}[[z^{\mp }]]^I$) if there exists a vector $m_0\in M$ such that

$$\begin{align} & e_{k,r}\ast m_0 = 0, \qquad {\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})^- \ast m_0 = M, \tag{1.2.14}\\ & q^h\ast m_0 = \varepsilon ^{\langle h, \Lambda \rangle } m_0 \quad \text{for $h\in P^*$}, \qquad \psi ^{\pm }_{k}(z)\ast m_0 = \Psi ^{\pm }_{k}(z) m_0 \quad \text{for $k\in I$}. \tag{1.2.15} \end{align}$$

By using 1.2.13 and a standard argument, one can show that there is a simple l-highest weight module $M$ of ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})$ with l-highest weight vector $m_0$ satisfying the above for any $(\Lambda ,(\Psi ^{\pm }_{k}(z))_k)$ with $\Psi ^+_k(\infty ) = (\alpha _k,\alpha _k)\langle \Lambda , h_k\rangle /2$, $\Psi ^-_k(0) = -(\alpha _k,\alpha _k)\langle \Lambda , h_k\rangle /2$. Moreover, such $M$ is unique up to isomorphism. For abuse of notation, we denote the pair $(\Lambda , (\Psi ^{\pm }_{k}(z))_k)$ simply by the symbol $\Psi ^\pm (z)$.

A ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})$-module $M$ is said to be l-integrable if

(a)

$M$ has a weight space decomposition $M = \bigoplus _{\lambda \in P} M_\lambda$ as a ${\mathbf{U}}_{\varepsilon }({\mathfrak{g}})$-module such that $\dim M_\lambda < \infty$,

(b)

for any $m\in M$, there exists $n_0\ge 1$ such that $e_{k,r_1}\cdots e_{k,r_n} \ast m = f_{k,r_1}\cdots f_{k,r_n} \ast m = 0$ for all $r_1$, …, $r_n\in {\mathbb{Z}}$, $k\in I$ and $n\ge n_0$.

For example, if ${\mathfrak{g}}$ is finite dimensional, and $M$ is a finite dimensional module, then $M$ satisfies the above conditions after twisting with a certain automorphism of ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})$ (10, 12.2.3).

Proposition 1.2.16.

Assume that ${\mathfrak{g}}$ is symmetric. The simple l-highest weight ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})$-module $M$ with l-highest weight $\Psi ^{\pm }(z)$ is l-integrable if and only if $\Lambda$ is dominant and there exist polynomials $P_k(u)\in {\mathbb{C}}[u]$ for $k\in I$ with $P_k(0) = 1$ such that

$$\begin{equation} \Psi ^{\pm }_{k}(z) = \varepsilon _k^{\deg P_k} \left(\frac{P_k(\varepsilon _k^{-1}/z)}{P_k(\varepsilon _k/z)}\right)^{\pm }, {\tag{1.2.17}} \end{equation}$$

where $\varepsilon _k = \varepsilon ^{(\alpha _k,\alpha _k)/2}$, and $\left(\ \right)^{\pm } \in {\mathbb{C}}[[z^{\mp }]]$ denotes the expansion at $z = \infty$ and $0$ respectively.

This result was announced by Drinfel’d for the Yangian 15. The proof of the ‘only if’ part when ${\mathfrak{g}}$ is finite dimensional was given by Chari-Pressley 10, 12.2.6. Since the proof is based on a reduction to the case ${\mathfrak{g}} = {\mathfrak{sl}}_2$, it can be applied to a general Kac-Moody algebra ${\mathfrak{g}}$ (not necessarily symmetric). The ‘if’ part was proved by them later in 11 when $\mathfrak{g}$ is finite dimensional, again not necessarily symmetric. As an application of the main result of this paper, we will prove the converse for a symmetric Kac-Moody algebra ${\mathfrak{g}}$ in §13. Our proof is independent of Chari-Pressley’s proof.

Remark 1.2.18.

The polynomials $P_k$ are called Drinfel’d polynomials.

When the Drinfel’d polynomials are given by

$$\begin{equation*} P_k(u) = \begin{cases} 1 - su & \text{if $k\neq k_0$,} \\ 1 & \text{otherwise}, \end{cases} \end{equation*}$$

for some $k_0\in I$, $s\in {\mathbb{C}}^*$, the corresponding simple l-highest weight module is called an l-fundamental representation. When ${\mathfrak{g}}$ is finite dimensional, ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})$ is a Hopf algebra since Drinfel’d 15 announced and Beck 5 proved that ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})$ can be identified with (a quotient of) the specialized quantized enveloping algebra associated with Cartan data of affine type. Thus a tensor product of ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})$-modules is again a ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})$-module. We have the following:

Proposition 1.2.19 (10, 12.2.6,12.2.8).

Suppose ${\mathfrak{g}}$ is finite dimensional.

(1) If $M$ and $N$ are simple l-highest weight ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})$-modules with Drinfel’d polynomials $P_{k,M}$, $P_{k,N}$ such that $M\otimes N$ is simple, then its Drinfel’d polynomial $P_{k,M\otimes N}$ is given by

$$\begin{equation*} P_{k,M\otimes N} = P_{k,M} P_{k,N}. \end{equation*}$$

(2) Every simple l-highest weight ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})$-module is a subquotient of a tensor product of l-fundamental representations.

Unfortunately the coproduct is not defined for general ${\mathfrak{g}}$ as far as the author knows. Thus the above results do not make sense for general ${\mathfrak{g}}$.

1.3. An l-weight space decomposition

Let $M$ be an l-integrable ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})$-module with the weight space decomposition $M = \bigoplus _{\lambda \in P} M_\lambda$. Since the commutative subalgebra ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})^0$ preserves each $M_\lambda$, we can further decompose $M$ into a sum of generalized simultaneous eigenspaces for ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})^0$:

$$\begin{equation} M = \bigoplus M_{\Psi ^\pm }, {\tag{1.3.1}} \end{equation}$$

where $\Psi ^\pm (z)$ is a pair $(\Lambda ,(\Psi _k^\pm (z))_k)$ as before and

$$\begin{equation*} M_{\Psi ^\pm } \overset{\operatorname {\scriptstyle def.}}{=}\left\{ m\in M \left|\, \begin{aligned} & \text{$q^h \ast m = \varepsilon ^{\langle h,\Lambda \rangle } m$  for $h\in P^*$}, \\ & \text{$(\psi _k^\pm (z) - \Psi _k^\pm (z)\operatorname {Id})^N \ast m = 0$} \\ &\qquad \qquad \qquad \quad \qquad \text{for $k\in I$ and sufficiently large $N$} \end{aligned} \right\}\right. . \end{equation*}$$

If $M_{\Psi ^\pm }\neq 0$, we call $M_{\Psi ^\pm }$ an l-weight space, and the corresponding $\Psi ^\pm (z)$ an l-weight. This is a refinement of the weight space decomposition. A further study of the l-weight space decomposition will be given in §13.5.

Motivated by Proposition 1.2.16, we introduce the following notion:

Definition 1.3.2.

An l-weight $\Psi ^\pm (z) = (\Lambda , (\Psi _k^\pm (z))_k)$ is said to be l-dominant if $\Lambda$ is dominant and there exists a polynomial $P(u) = (P_k(u))_k\in {\mathbb{C}}[u]^I$ for with $P_k(0) = 1$ such that 1.2.17 holds.

Thus Proposition 1.2.16 means that an l-highest weight module is l-integrable if and only if the l-highest weight is l-dominant.

2. Quiver variety

2.1. Notation

Suppose that a finite graph is given and assume that there are no edge loops, i.e., no edges joining a vertex with itself. Let $I$ be the set of vertices and $E$ the set of edges. Let ${\mathbf{A}}$ be the adjacency matrix of the graph, namely

$$\begin{equation*} {\mathbf{A}}= ({\mathbf{A}}_{kl})_{k,l\in I}, \qquad \text{where ${\mathbf{A}}_{kl}$ is the number of edges joining $k$ and $l$}. \end{equation*}$$

We associate with the graph $(I, E)$ a symmetric generalized Cartan matrix ${\mathbf{C}}= 2{\mathbf{I}}- {\mathbf{A}}$, where ${\mathbf{I}}$ is the identity matrix. This gives a bijection between the finite graphs without edge loops and symmetric Cartan matrices. We have the corresponding symmetric Kac-Moody algebra $\mathfrak{g}$, the quantized enveloping algebra ${\mathbf{U}}_q({\mathfrak{g})}$, the quantum affine algebra ${\mathbf{U}}_q(\widehat{\mathfrak{g}})$ and the quantum loop algebra ${\mathbf{U}}_q({\mathbf{L}}{\mathfrak{g}})$. Let $H$ be the set of pairs consisting of an edge together with its orientation. For $h\in H$, we denote by $\operatorname {in}(h)$ (resp. $\operatorname {out}(h)$) the incoming (resp. outgoing) vertex of $h$. For $h\in H$ we denote by $\overline{h}$ the same edge as $h$ with the reverse orientation. Choose and fix an orientation $\Omega$ of the graph, i.e., a subset $\Omega \subset H$ such that $\overline{\Omega }\cup \Omega = H$, $\Omega \cap \overline{\Omega }= \emptyset$. The pair $(I,\Omega )$ is called a quiver. Let us define matrices ${\mathbf{A}}_\Omega$ and ${\mathbf{A}}_{\overline{\Omega }}$ by

$$\begin{equation} \begin{split} ({\mathbf{A}}_\Omega )_{kl} & \overset{\operatorname {\scriptstyle def.}}{=}\# \{ h\in \Omega \mid \operatorname {in}(h) = k, \operatorname {out}(h) = l\}, \\ ({\mathbf{A}}_{\overline{\Omega }})_{kl} & \overset{\operatorname {\scriptstyle def.}}{=}\# \{ h\in \overline{\Omega } \mid \operatorname {in}(h) = k, \operatorname {out}(h) = l\}. \end{split} {\tag{2.1.1}} \end{equation}$$

So we have ${\mathbf{A}}= {\mathbf{A}}_\Omega + {\mathbf{A}}_{\overline{\Omega }}$, $\,{}^{t}{{\mathbf{A}}}_\Omega = {\mathbf{A}}_{\overline{\Omega }}$.

Let $V = (V_k)_{k\in I}$ be a collection of finite-dimensional vector spaces over ${\mathbb{C}}$ for each vertex $k\in I$. The dimension of $V$ is a vector

$$\begin{equation*} \dim V = (\dim V_k)_{k\in I}\in \mathbb{Z}_{\ge 0}^I. \end{equation*}$$

If $V^1$ and $V^2$ are such collections, we define vector spaces by

$$\begin{equation} \begin{split} & \operatorname {L}(V^1, V^2) \overset{\operatorname {\scriptstyle def.}}{=}\bigoplus _{k\in I} \operatorname {Hom}(V^1_k, V^2_k), \\ & \operatorname {E}(V^1, V^2) \overset{\operatorname {\scriptstyle def.}}{=}\bigoplus _{h\in H} \operatorname {Hom}(V^1_{\operatorname {out}(h)}, V^2_{\operatorname {in}(h)}). \end{split}{\tag{2.1.2}} \end{equation}$$

For $B = (B_h) \in \operatorname {E}(V^1, V^2)$ and $C = (C_h) \in \operatorname {E}(V^2, V^3)$, let us define a multiplication of $B$ and $C$ by

$$\begin{equation*} CB \overset{\operatorname {\scriptstyle def.}}{=}(\sum _{\operatorname {in}(h) = k} C_h B_{\overline{h}})_k \in \operatorname {L}(V^1, V^3). \end{equation*}$$

Multiplications $ba$, $Ba$ of $a\in \operatorname {L}(V^1,V^2)$, $b\in \operatorname {L}(V^2, V^3)$, $B\in \operatorname {E}(V^2, V^3)$ are defined in an obvious manner. If $a\in \operatorname {L}(V^1, V^1)$, its trace $\operatorname {tr}(a)$ is understood as $\sum _k \operatorname {tr}(a_k)$.

For two collections $V$, $W$ of vector spaces with ${\mathbf{v}}= \dim V$, ${\mathbf{w}}= \dim W$, we consider the vector space given by

$$\begin{equation} {\mathbf{M}}\equiv {\mathbf{M}}({\mathbf{v}}, {\mathbf{w}}) \overset{\operatorname {\scriptstyle def.}}{=}\operatorname {E}(V, V) \oplus \operatorname {L}(W, V) \oplus \operatorname {L}(V, W), {\tag{2.1.3}} \end{equation}$$

where we use the notation ${\mathbf{M}}$ unless we want to specify dimensions of $V$, $W$. The above three components for an element of ${\mathbf{M}}$ will be denoted by $B$, $i$, $j$ respectively. An element of ${\mathbf{M}}$ will be called an ADHM datum.

Usually a point in $\bigoplus _{h\in \Omega } \operatorname {Hom}(V^1_{\operatorname {out}(h)}, V^2_{\operatorname {in}(h)})$ is called a representation of the quiver $(I,\Omega )$ in the literature. Thus $\operatorname {E}(V,V)$ is the product of the space of representations of $(I,\Omega )$ and that of $(I,\overline{\Omega })$. On the other hand, the factor $\operatorname {L}(W,V)$ or $\operatorname {L}(V,W)$ has never appeared in the literature.

Convention 2.1.4.

When we relate the quiver varieties to the quantum affine algebra, the dimension vectors will be mapped into the weight lattice in the following way:

$$\begin{equation*} {\mathbf{v}}\mapsto \sum _k v_k \alpha _k, \quad {\mathbf{w}}\mapsto \sum _k w_k \Lambda _k, \end{equation*}$$

where $v_k$ (resp. $w_k$) is the $k$th component of ${\mathbf{v}}$ (resp. ${\mathbf{w}}$). Since $\{ \alpha _k \}$ and $\{ \Lambda _k \}$ are both linearly independent, these maps are injective. We consider ${\mathbf{v}}$ and ${\mathbf{w}}$ as elements of the weight lattice $P$ in this way hereafter.

For a collection $S = (S_k)_{k\in I}$ of subspaces of $V_k$ and $B\in \operatorname {E}(V, V)$, we say $S$ is $B$-invariant if $B_h(S_{\operatorname {out}(h)}) \subset S_{\operatorname {in}(h)}$.

Fix a function $\varepsilon \colon H \to {\mathbb{C}}^*$ such that $\varepsilon (h) + \varepsilon (\overline{h}) = 0$ for all $h\in H$. In 4445, it was assumed that $\varepsilon$ takes its value $\pm 1$, but this assumption is not necessary as remarked by Lusztig 38. For $B\in \operatorname {E}(V^1, V^2)$, let us denote by $\varepsilon B\in \operatorname {E}(V^1, V^2)$ data given by $(\varepsilon B)_h = \varepsilon (h) B_h$ for $h\in H$.

Let us define a symplectic form $\omega$ on ${\mathbf{M}}$ by

$$\begin{equation} \omega ((B, i, j), (B', i', j')) \overset{\operatorname {\scriptstyle def.}}{=}\operatorname {tr}(\varepsilon B\, B') + \operatorname {tr}(i j' - i' j). {\tag{2.1.5}} \end{equation}$$

Let $G$ be the algebraic group defined by

$$\begin{equation*} G \equiv G_{{\mathbf{v}}} \overset{\operatorname {\scriptstyle def.}}{=}\prod _k \operatorname {GL}(V_k), \end{equation*}$$

where we use the notation $G_{\mathbf{v}}$ when we want to emphasize the dimension. It acts on ${\mathbf{M}}$ by

$$\begin{equation} (B, i, j) \mapsto g\cdot (B, i, j) \overset{\operatorname {\scriptstyle def.}}{=}(g B g^{-1}, g i, j g^{-1}), {\tag{2.1.6}} \end{equation}$$

preserving the symplectic form $\omega$. The moment map $\mu \colon {\mathbf{M}}\to \operatorname {L}(V, V)$ vanishing at the origin is given by

$$\begin{equation} \mu (B, i, j) = \varepsilon B B + i j, {\tag{2.1.7}} \end{equation}$$

where the dual of the Lie algebra of $G$ is identified with the Lie algebra via the trace. Let $\mu ^{-1}(0)$ be an affine algebraic variety (not necessarily irreducible) defined as the zero set of $\mu$.

For $(B,i,j)\in \mu ^{-1}(0)$, we consider the complex

$$\begin{equation} \operatorname {L}(V, V) \overset{\iota }{\longrightarrow } \operatorname {E}(V, V) \oplus \operatorname {L}(W, V) \oplus \operatorname {L}(V,W) \overset{d \mu }{\longrightarrow } \operatorname {L}(V, V), {\tag{2.1.8}} \end{equation}$$

where $d\mu$ is the differential of $\mu$ at $(B,i,j)$, and $\iota$ is given by

$$\begin{equation*} \iota (\xi ) = (B \xi - \xi B) \oplus (-\xi i) \oplus j \xi . \end{equation*}$$

If we identify $\operatorname {E}(V, V) \oplus \operatorname {L}(W, V) \oplus \operatorname {L}(V,W)$ with its dual via the symplectic form $\omega$, $\iota$ is the transpose of $d \mu$.

2.2. Two quotients ${\mathfrak{M}}_0$ and ${\mathfrak{M}}$

We consider two types of quotients of $\mu ^{-1}(0)$ by the group $G$. The first one is the affine algebro-geometric quotient given as follows. Let $A(\mu ^{-1}(0))$ be the coordinate ring of the affine algebraic variety $\mu ^{-1}(0)$. Then ${\mathfrak{M}}_0$ is defined as a variety whose coordinate ring is the invariant part of $A(\mu ^{-1}(0))$:

$$\begin{equation} {\mathfrak{M}}_0 \equiv {\mathfrak{M}}_0({\mathbf{v}},{\mathbf{w}}) \overset{\operatorname {\scriptstyle def.}}{=}\mu ^{-1}(0)\mathord{\sslash }G = \operatorname {Spec}\nolimits A(\mu ^{-1}(0))^{G}. \tag{2.2.1} \end{equation}$$

As before, we use the notation ${\mathfrak{M}}_0$ unless we need to specify the dimension vectors ${\mathbf{v}}$, ${\mathbf{w}}$. By the geometric invariant theory 43, this is an affine algebraic variety. It is also known that the geometric points of ${\mathfrak{M}}_0$ are closed $G$-orbits.

For the second quotient we follow A. King’s approach 27. Let us define a character $\chi \colon G\to {\mathbb{C}}^*$ by $\chi (g) = \prod _k \det g_k^{-1}$ for $g = (g_k)$. Set

$$\begin{equation*} A(\mu ^{-1}(0))^{G, \chi ^n} \overset{\operatorname {\scriptstyle def.}}{=}\{\, f\in A(\mu ^{-1}(0)) \mid f(g (B, i, j)) = \chi (g)^n f(B, i, j) \,\} . \end{equation*}$$

The direct sum with respect to $n\in \mathbb{Z}_{\ge 0}$ is a graded algebra, hence we can define

$$\begin{equation} {\mathfrak{M}}\equiv {\mathfrak{M}}({\mathbf{v}},{\mathbf{w}}) \overset{\operatorname {\scriptstyle def.}}{=}\operatorname {Proj}\nolimits \bigoplus _{n\ge 0} A(\mu ^{-1}(0))^{G, \chi ^n}. \tag{2.2.2} \end{equation}$$

These are what we call quiver varieties.

2.3. Stability condition

In this subsection, we shall give a description of the quiver variety ${\mathfrak{M}}$ which is easier to deal with. We again follow King’s work 27.

Definition 2.3.1.

A point $(B, i, j) \in \mu ^{-1}(0)$ is said to be stable if the following condition holds:

if a collection $S = (S_k)_{k\in I}$ of subspaces of $V = (V_k)_{k\in I}$ is $B$-invariant and contained in $\operatorname {Ker}j$, then $S = 0$.

Let us denote by $\mu ^{-1}(0)^{\operatorname {s}}$ the set of stable points.

Clearly, the stability condition is invariant under the action of $G$. Hence we may say an orbit is stable or not.

Let us lift the $G$-action on $\mu ^{-1}(0)$ to the trivial line bundle $\mu ^{-1}(0) \times {\mathbb{C}}$ by $g\cdot (B,i,j,z) = (g\cdot (B,i,j), \chi ^{-1}(g)z)$.

We have the following:

Proposition 2.3.2.

(1) A point $(B,i,j)$ is stable if and only if the closure of $G\cdot (B,i,j,z)$ does not intersect with the zero section of $\mu ^{-1}(0) \times {\mathbb{C}}$ for $z\neq 0$.

(2) If $(B,i,j)$ is stable, then the differential $d\mu \colon {\mathbf{M}}\to \operatorname {L}(V,V)$ is surjective. In particular, $\mu ^{-1}(0)^{\operatorname {s}}$ is a nonsingular variety.

(3) If $(B,i,j)$ is stable, then $\iota$ in 2.1.8 is injective.

(4) The quotient $\mu ^{-1}(0)^{\operatorname {s}}/G$ has a structure of nonsingular quasi-projective variety of dimension $({\mathbf{v}}, 2{\mathbf{w}}- {\mathbf{v}})$, and $\mu ^{-1}(0)^{\operatorname {s}}$ is a principal $G$-bundle over $\mu ^{-1}(0)^{\operatorname {s}}/G$.

(5) The tangent space of $\mu ^{-1}(0)^{\operatorname {s}}/G$ at the orbit $G\cdot (B,i,j)$ is isomorphic to the middle cohomology group of 2.1.8.

(6) The variety ${\mathfrak{M}}$ is isomorphic to $\mu ^{-1}(0)^{\operatorname {s}}/G$.

(7) $\mu ^{-1}(0)^{\operatorname {s}}/G$ has a holomorphic symplectic structure as a symplectic quotient.

Proof.

See 45, 3.ii and 44, 2.8.

■

Notation 2.3.3.

For a stable point $(B, i, j)\in \mu ^{-1}(0)$, its $G$-orbit considered as a geometric point in the quiver variety ${\mathfrak{M}}$ is denoted by $[B, i, j]$. If $(B, i, j)\in \mu ^{-1}(0)$ has a closed $G$-orbit, then the corresponding geometric point in ${\mathfrak{M}}_0$ will be denoted also by $[B, i, j]$.

From the definition, we have a natural projective morphism (see 45, 3.18)

$$\begin{equation} \pi \colon {\mathfrak{M}}\to {\mathfrak{M}}_0. {\tag{2.3.4}} \end{equation}$$

If $\pi ([B,i,j]) = [B^0,i^0,j^0]$, then $G\cdot (B^0, i^0, j^0)$ is the unique closed orbit contained in the closure of $G\cdot (B, i, j)$. For $x\in {\mathfrak{M}}_0$, let

$$\begin{equation} {\mathfrak{M}}_x \overset{\operatorname {\scriptstyle def.}}{=}\pi ^{-1}(x). {\tag{2.3.5}} \end{equation}$$

If we want to specify the dimension, we denote the above by ${\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})_x$. Unfortunately, this notation conflicts with the previous notation ${\mathfrak{M}}_0$ when $x = 0$. And the central fiber $\pi ^{-1}(0)$ plays an important role later. We shall always write ${\mathfrak{L}}\equiv {\mathfrak{L}}({\mathbf{v}},{\mathbf{w}})$ for $\pi ^{-1}(0)$ and not use the notation 2.3.5 with $x = 0$.

In order to explain a more precise relation between $[B,i,j]$ and $[B^0,i^0,j^0]$, we need the following notion.

Definition 2.3.6.

Suppose that $(B, i, j)\in {\mathbf{M}}$ and a $B$-invariant increasing filtration

$$\begin{equation*} 0 = V^{(-1)} \subset V^{(0)} \subset \cdots \subset V^{(N)} = V \end{equation*}$$

with $\operatorname {Im}i \subset V^{(0)}$ are given. Then set $\operatorname {gr}_m V = V^{(m)}/V^{(m-1)}$ and $\operatorname {gr}V = \bigoplus \operatorname {gr}_m V$. Let $\operatorname {gr}_m B$ denote the endomorphism which $B$ induces on $\operatorname {gr}_m V$. For $m = 0$, let $\operatorname {gr}_0 i\in \operatorname {L}(W, V^{(0)})$ be such that its composition with the inclusion $V^{(0)} \subset V$ is $i$, and let $\operatorname {gr}_0 j$ be the restriction of $j$ to $V^{(0)}$. For $m \ne 0$, set $\operatorname {gr}_m i = 0$ and $\operatorname {gr}_m j = 0$. Let $\operatorname {gr}(B, i, j)$ be the direct sum of $(\operatorname {gr}_m B, \operatorname {gr}_m i, \operatorname {gr}_m j)$ considered as data on $\operatorname {gr}V$.

Proposition 2.3.7.

Suppose $\pi (x) = y$. Then there exist a representative $(B, i, j)$ of $x$ and a $B$-invariant increasing filtration $V^{(*)}$ as in Definition 2.3.6 such that $\operatorname {gr}(B, i, j)$ is a representative of $y$ on $\operatorname {gr}V$.

Proof.

See 45, 3.20

■

Proposition 2.3.8.

${\mathfrak{L}}$ is a Lagrangian subvariety which is homotopic to ${\mathfrak{M}}$.

Proof.

See 44, 5.5, 5.8.

■

2.4. Hyper-Kähler structure

We briefly recall hyper-Kähler structures on ${\mathfrak{M}}$, ${\mathfrak{M}}_0$. This viewpoint was used for the study of ${\mathfrak{M}}$, ${\mathfrak{M}}_0$ in 44. (Caution: The following notation is different from the original one. $K_{\mathbf{v}}$ and $G_{\mathbf{v}}$ were denoted by $G_{\mathbf{v}}$ and $G_{\mathbf{v}}^{\mathbb{C}}$ respectively in 44. $\mu$ in 2.1.7 was denoted by $\mu _{\mathbb{C}}$ and the pair $(\mu _{\mathbb{R}},\mu )$ was denoted by $\mu$ in 44.)

Put and fix hermitian inner products on $V$ and $W$. They together with an orientation $\Omega$ induce a hermitian inner product and a quaternion structure on ${\mathbf{M}}$ (44, p.370). Let $K_{\mathbf{v}}$ be a compact Lie group defined by $K_{\mathbf{v}}= \prod _k \operatorname {U}(V_k)$. This is a maximal compact subgroup of $G_{\mathbf{v}}$, and acts on ${\mathbf{M}}$ preserving the hermitian and quaternion structures. The corresponding hyper-Kähler moment map vanishing at the origin decomposes into the complex part $\mu \colon {\mathbf{M}}\to \bigoplus _k {\mathfrak{gl}}(V_k) = \operatorname {L}(V,V)$ (defined in 2.1.7) and the real part $\mu _{\mathbb{R}}\colon {\mathbf{M}}\to \bigoplus _k {\mathfrak{u}}(V_k)$, where

$$\begin{equation*} \mu _{\mathbb{R}}(B,i,j) = {i\over 2}\left( \sum _{h\in H: k = \operatorname {in}(h)} B_h B_h^\dagger - B_{\overline{h}}^\dagger B_{\overline{h}} + i_k i_k^\dagger - j_k^\dagger j_k\right)_k . \end{equation*}$$

Proposition 2.4.1.

(1) A $G_{\mathbf{v}}$-orbit $[B,i,j]$ in $\mu ^{-1}(0)$ intersects with $\mu _{\mathbb{R}}^{-1}(0)$ if and only if it is closed. The map

$$\begin{equation*} \left(\mu _{\mathbb{R}}^{-1}(0)\cap \mu ^{-1}(0)\right) / K_{\mathbf{v}}\to \mu ^{-1}(0)\mathord{\sslash }G_{\mathbf{v}}= {\mathfrak{M}}_0({\mathbf{v}},{\mathbf{w}}) \end{equation*}$$

is a homeomorphism.

(2) Choose a parameter $\zeta _{\mathbb{R}}= (\zeta _{\mathbb{R}}^{(k)})_k \in {\mathbb{R}}^I$ so that $\zeta _{\mathbb{R}}^{(k)}\in \sqrt {-1}{\mathbb{R}}_{> 0}$. Then a $G_{\mathbf{v}}$-orbit $[B,i,j]$ in $\mu ^{-1}(0)$ intersects with $\mu _{\mathbb{R}}^{-1}(-\zeta _{\mathbb{R}})$ if and only if it is stable. The map

$$\begin{equation*} \left(\mu _{\mathbb{R}}^{-1}(-\zeta _{\mathbb{R}})\cap \mu ^{-1}(0)\right) / K_{\mathbf{v}}\to \mu ^{-1}(0)^{\operatorname {s}}/G = {\mathfrak{M}}({\mathbf{v}},{\mathbf{w}}) \end{equation*}$$

is a homeomorphism.

Proof.

See 44, 3.1,3.2,3.5.

■

2.5

Suppose $V = (V_k)_{k\in I}$ is a collection of subspaces of $V' = (V'_k)_{k\in I}$ and $(B,i,j)\in \mu ^{-1}(0)\subset {\mathbf{M}}({\mathbf{v}},{\mathbf{w}})$ is given. We can extend $(B,i,j)$ to ${\mathbf{M}}({\mathbf{v}}',{\mathbf{w}})$ by letting it equal $0$ on a complementary subspace of $V$ in $V'$. This operation induces a natural morphism

$$\begin{equation} \text{$\mu ^{-1}(0)$ in ${\mathbf{M}}({\mathbf{v}},{\mathbf{w}})$} \to \text{$\mu ^{-1}(0)$ in ${\mathbf{M}}({\mathbf{v}}',{\mathbf{w}})$}, {\tag{2.5.1}} \end{equation}$$

where ${\mathbf{v}}' = \dim V'$. This induces a morphism

$$\begin{equation} {\mathfrak{M}}_0({\mathbf{v}},{\mathbf{w}}) \to {\mathfrak{M}}_0({\mathbf{v}}',{\mathbf{w}}). {\tag{2.5.2}} \end{equation}$$

Moreover, we also have a map

$$\begin{equation*} \text{$\mu ^{-1}(0)\cap \mu _{\mathbb{R}}^{-1}(0)$ in ${\mathbf{M}}({\mathbf{v}},{\mathbf{w}})$} \to \text{$\mu ^{-1}(0)\cap \mu _{\mathbb{R}}^{-1}(0)$ in ${\mathbf{M}}({\mathbf{v}}',{\mathbf{w}})$}. \end{equation*}$$

Thus closed $G_{\mathbf{v}}$-orbits in $\mu ^{-1}(0)\subset {\mathbf{M}}({\mathbf{v}},{\mathbf{w}})$ are mapped to closed $G_{{\mathbf{v}}'}$-orbits in $\mu ^{-1}(0)\subset {\mathbf{M}}({\mathbf{v}}',{\mathbf{w}})$ by Proposition 2.4.1(1).

The following lemma was stated in 45, p.529 without proof.

Lemma 2.5.3.

The morphism 2.5.2 is injective.

Proof.

Suppose $x^1$, $x^2\in {\mathfrak{M}}_0({\mathbf{v}},{\mathbf{w}})$ have the same image under 2.5.2. We choose representatives $(B^1,i^1,j^1)$, $(B^2,i^2,j^2)$ which have closed $G_{{\mathbf{v}}}$-orbits.

Let us define $S^a = (S^a_k)_{k\in I}$ ($a = 1,2$) by

$$\begin{equation*} S^a_k \overset{\operatorname {\scriptstyle def.}}{=}\operatorname {Im}\left(\sum _{\operatorname {in}(h)=k} \varepsilon (h) B^a_h + i^a_k\right). \end{equation*}$$

Choose complementary subspaces $T_k^a$ of $S^a_k$ in $V_k$. We choose a $1$-parameter subgroup $\lambda ^a\colon {\mathbb{C}}^*\to G_{{\mathbf{v}}}$ as follows: $\lambda ^a(t) = 1$ on $S^a_k$ and $\lambda ^a(t) = t^{-1}$ on $T_k^a$. Then the limit $\lambda ^a(t)\cdot (B^a,i^a,j^a)$ exists and its restriction to $T_k^a$ is $0$. Since $(B^a,i^a,j^a)$ has a closed orbit, we may assume that the restriction of $(B^a,i^a,j^a)$ to $T_k^a$ is $0$. Note that $S^a$ is a subspace of $V$ by the construction.

Suppose that there exists $g'\in G_{{\mathbf{v}}'}$ such that $g'\cdot (B^1,i^1,j^1) = (B^2,i^2,j^2)$. We want to construct $g\in G_{\mathbf{v}}$ such that $g\cdot (B^1,i^1,j^1) = (B^2,i^2,j^2)$. Since we have $g'(S^1) = S^2$, the restriction of $g'$ to $S^1$ is invertible. Let $g$ be an extension of the restriction $g'|_{S^1}$ to $V$ so that $T^1$ is mapped to $T^2$. Then $g\in G_{\mathbf{v}}$ maps $(B^1,i^1,j^1)$ to $(B^2,i^2,j^2)$.

■

Hereafter, we consider ${\mathfrak{M}}_0({\mathbf{v}},{\mathbf{w}})$ as a subset of ${\mathfrak{M}}_0({\mathbf{v}}',{\mathbf{w}})$. It is clearly a closed subvariety. Let

$$\begin{equation} {\mathfrak{M}}_0(\infty ,{\mathbf{w}}) \overset{\operatorname {\scriptstyle def.}}{=}\bigcup _{{\mathbf{v}}} {\mathfrak{M}}_0({\mathbf{v}},{\mathbf{w}}). {\tag{2.5.4}} \end{equation}$$

If the graph is of finite type, ${\mathfrak{M}}_0({\mathbf{v}},{\mathbf{w}})$ stabilizes at some ${\mathbf{v}}$ (see Proposition 2.6.3 and Lemma 2.9.4(2) below). This is not true in general. However, it presents no harm in this paper. We use ${\mathfrak{M}}_0(\infty ,{\mathbf{w}})$ to simplify the notation, and do not need any structures on it. We can always work on individual ${\mathfrak{M}}_0({\mathbf{v}},{\mathbf{w}})$, not on ${\mathfrak{M}}_0(\infty ,{\mathbf{w}})$.

Later, we shall also study ${\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})$ for various ${\mathbf{v}}$ simultaneously. We introduce the following notation:

$$\begin{equation*} {\mathfrak{M}}({\mathbf{w}})\overset{\operatorname {\scriptstyle def.}}{=}\bigsqcup _{{\mathbf{v}}}{\mathfrak{M}}({\mathbf{v}},{\mathbf{w}}), \qquad {\mathfrak{L}}({\mathbf{w}})\overset{\operatorname {\scriptstyle def.}}{=}\bigsqcup _{{\mathbf{v}}}{\mathfrak{L}}({\mathbf{v}},{\mathbf{w}}). \end{equation*}$$

Note that there are no obvious morphisms between ${\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})$ and ${\mathfrak{M}}({\mathbf{v}}',{\mathbf{w}})$ since the stability condition is not preserved under 2.5.1.

2.6. Definition of ${\mathfrak{M}_0^{\operatorname {reg}}}$

Let us introduce an open subset of ${\mathfrak{M}}_0$ (possibly empty):

$$\begin{equation} \begin{split} {\mathfrak{M}_0^{\operatorname {reg}}}&\equiv {\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}})\\ & \overset{\operatorname {\scriptstyle def.}}{=}\{\, [B, i, j]\in {\mathfrak{M}}_0\mid \text{$(B, i, j)$ has the trivial stabilizer in $G$}\,\}. \end{split}{\tag{2.6.1}} \end{equation}$$

Proposition 2.6.2.

If $[B, i, j]\in {\mathfrak{M}_0^{\operatorname {reg}}}$, then it is stable. Moreover, $\pi$ induces an isomorphism $\pi ^{-1}({\mathfrak{M}_0^{\operatorname {reg}}})\simeq {\mathfrak{M}_0^{\operatorname {reg}}}$.

Proof.

See 45, 3.24 or 44, 4.1(2).

■

As in §2.5, we consider ${\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}})$ as a subset of ${\mathfrak{M}}_0({\mathbf{v}}',{\mathbf{w}})$ when ${\mathbf{v}}' - {\mathbf{v}}\in \sum _k {\mathbb{Z}}_{\ge 0} \alpha _k$. Then we have

Proposition 2.6.3.

If the graph is of type $ADE$, then

$$\begin{equation*} {\mathfrak{M}}_0({\mathbf{v}}',{\mathbf{w}}) = \bigcup _{{\mathbf{v}}} {\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}}), \end{equation*}$$

where the summation runs over the set of ${\mathbf{v}}$ such that ${\mathbf{v}}' - {\mathbf{v}}\in \sum _k {\mathbb{Z}}_{\ge 0}\alpha _k$.

Proof.

See 44, 6.7, 45, 3.28.

■

Definition 2.6.4.

We say a point $x\in {\mathfrak{M}}_0(\infty ,{\mathbf{w}})$ is regular if it is contained in ${\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}})$ for some ${\mathbf{v}}$. The above proposition says that all points are regular if the graph is of type $ADE$. But this is not true in general (see 45, 10.10).

2.7. $G_{\mathbf{w}}\times {\mathbb{C}}^*$-action

Let us define a $G_{\mathbf{w}}\times {\mathbb{C}}^*$-action on ${\mathfrak{M}}$ and ${\mathfrak{M}}_0$, where $G_{\mathbf{w}}= \prod _{k\in I} \operatorname {GL}(W_k)$. (Caution: We use the same notation $G_{\mathbf{v}}$ and $G_{\mathbf{w}}$, but their roles are totally different.)

The $G_{\mathbf{w}}$-action is simply defined by its natural action on ${\mathbf{M}}= \operatorname {E}(V,V)\oplus \operatorname {L}(W,V)\oplus \operatorname {L}(V,W)$. It preserves the equation $\varepsilon B B + ij = 0$ and commutes with the $G$-action given by 2.1.6. Hence it induces an action on ${\mathfrak{M}}$ and ${\mathfrak{M}}_0$.

The ${\mathbb{C}}^*$-action is slightly different from the one given in 45, 3.iv, and we need extra data. For each pair $k,l\in I$ such that $b' = -(\alpha _k,\alpha _l) \ge 1$, we introduce and fix a numbering $1$, $2$, …, $b'$ on edges joining $k$ and $l$. It induces a numbering $h_1$, …, $h_{b'}\in \Omega$, $\overline{h_1}$, …, $\overline{h_{b'}}\in \overline{\Omega }$ on oriented edges between $k$ and $l$. Let us define $m\colon H\to {\mathbb{Z}}$ by

$$\begin{equation} m(h_p) = b'+1-2p, \quad m(\overline{h}_p) = -b'-1+2p. {\tag{2.7.1}} \end{equation}$$

Then we define a ${\mathbb{C}}^*$-action on ${\mathbf{M}}$ by

$$\begin{equation} B_{h} \mapsto t^{m(h)+1} B_{h}, \quad i \mapsto t i, \quad j \mapsto t j \qquad \text{for $t\in {\mathbb{C}}^*$}. {\tag{2.7.2}} \end{equation}$$

The equation $\varepsilon B B + ij = 0$ is preserved since the left hand side is multiplied by $t^2$. It commutes with the $G$-action and preserves the stability condition. Hence it induces a ${\mathbb{C}}^*$-action on ${\mathfrak{M}}$ and ${\mathfrak{M}}_0$. This $G_{\mathbf{w}}\times {\mathbb{C}}^*$-action makes the projective morphism $\pi \colon {\mathfrak{M}}\to {\mathfrak{M}}_0$ equivariant.

In order to distinguish this $G_{\mathbf{w}}\times {\mathbb{C}}^*$-action from the $G_{\mathbf{v}}$-action 2.1.6, we denote it as

$$\begin{equation*} (B,i,j) \mapsto h\star (B,i,j) \qquad (h\in G_{\mathbf{w}}\times {\mathbb{C}}^*). \end{equation*}$$

2.8. Notation for ${\mathbb{C}}^*$-action

For an integer $m$, we define a ${\mathbb{C}}^*$-module structure on ${\mathbb{C}}$ by

$$\begin{equation} t\cdot v \overset{\operatorname {\scriptstyle def.}}{=}t^m v, \qquad \text{$t\in {\mathbb{C}}^*, v\in {\mathbb{C}}$}, {\tag{2.8.1}} \end{equation}$$

and denote it by $L(m)$. For a ${\mathbb{C}}^*$-module $V$, we use the following notational convention:

$$\begin{equation} q^m V \overset{\operatorname {\scriptstyle def.}}{=}L(m)\otimes V. {\tag{2.8.2}} \end{equation}$$

We use the same notation later when $V$ is an element of ${\mathbb{C}}^*$-equivariant $K$-theory.

2.9. Tautological bundles

By the construction of ${\mathfrak{M}}$, we have a natural vector bundle

$$\begin{equation*} \mu ^{-1}(0)^{\operatorname {s}}\times _G V_k \to {\mathfrak{M}} \end{equation*}$$

associated with the principal $G$-bundle $\mu ^{-1}(0)^{\operatorname {s}}\to {\mathfrak{M}}$. For abuse of notation, we denote it also by $V_k$. It naturally has the structure of a ${\mathbb{C}}^*$-equivariant vector bundle. Letting $G_{\mathbf{w}}$ act trivially, we make it a $G_{\mathbf{w}}\times {\mathbb{C}}^*$-equivariant vector bundle.

The vector space $W_k$ is also considered as a $G_{\mathbf{w}}\times {\mathbb{C}}^*$-equivariant vector bundle, where $G_{\mathbf{w}}$ acts naturally and ${\mathbb{C}}^*$ acts trivially.

We call $V_k$ and $W_k$ tautological bundles.

We consider $\operatorname {E}(V,V)$, $\operatorname {L}(W,V)$, $\operatorname {L}(V, W)$ as vector bundles defined by the same formula as in 2.1.2. By the definition of tautological bundles, $B$, $i$, $j$ can be considered as sections of those bundles. Those bundles naturally have structures of $G_{\mathbf{w}}\times {\mathbb{C}}^*$-equivariant vector bundles. But we modify the ${\mathbb{C}}^*$-action on $\operatorname {E}(V,V)$ by letting $t\in {\mathbb{C}}^*$ act by $t^{m(h)+1}$ on the component $\operatorname {Hom}(V_{\operatorname {out}(h)}, V_{\operatorname {in}(h)})$. This makes $B$ an equivariant section of $\operatorname {E}(V,V)$.

We consider the following $G_{\mathbf{w}}\times {\mathbb{C}}^*$-equivariant complex $C_k^\bullet \equiv C_k^\bullet ({\mathbf{v}},{\mathbf{w}})$ over ${\mathfrak{M}}\equiv {\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})$ (cf. 45, 4.2):

$$\begin{equation} C_k^\bullet \equiv C_k^\bullet ({\mathbf{v}},{\mathbf{w}}): \begin{CD} q^{-2} V_k @>{\sigma _k}>> q^{-1} \left( \displaystyle {\bigoplus _{l:k\neq l}} [-\langle h_k,\alpha _l\rangle ]_q V_l \oplus W_k\right) @>{\tau _k}>> V_k, \end{CD} {\tag{2.9.1}} \end{equation}$$

where

$$\begin{equation*} \sigma _k = \bigoplus _{\operatorname {in}(h)=k} B_{\overline{h}} \oplus j_k, \qquad \tau _k = \sum _{\operatorname {in}(h)=k} \varepsilon (h) B_h + i_k. \end{equation*}$$

Let us explain the factor $[-\langle h_k,\alpha _l\rangle ]_q V_l$. Set $b' = -\langle h_k,\alpha _l\rangle$. Since the ${\mathbb{C}}^*$-action in 2.7.2 is defined so that

$$\begin{equation*} \bigoplus _{h:\substack{\operatorname {in}(h) = k\\ \operatorname {out}(h)= l}} \operatorname {Hom}(V_k, V_l) = \operatorname {Hom}(V_k,V_l)^{\oplus b'} \end{equation*}$$

has weights $b'$, $b'-2$, …, $2 - b'$, the above can be written as

$$\begin{equation*} \left(q^{b'} + q^{b'-2} + \dots + q^{2-b'}\right) \operatorname {Hom}(V_k,V_l) = q[b']_q \operatorname {Hom}(V_k,V_l) \end{equation*}$$

in the notation 2.8.2. By the same reason $C_k^\bullet$ is an equivariant complex.

We assign degree $0$ to the middle term. (This complex is the complex in 45, 4.2 with a modification of the $G_{\mathbf{w}}\times {\mathbb{C}}^*$-action.)

Lemma 2.9.2.

Fix a point $[B,i,j]$ and consider $C_k^\bullet$ as a complex of vector spaces. Then $\sigma _k$ is injective.

Proof.

See 45, p.530. (Lemma 54 therein is a misprint of Lemma 5.2.)

■

Note that $\tau _k$ is not surjective in general. In fact, the following notion will play a crucial role later. Let $X$ be an irreducible component of $\pi ^{-1}(x)$ for $x\in {\mathfrak{M}}_0$. Considering $\tau _k$ at a generic element $[B,i,j]$ of $X$, we set

$$\begin{equation} \varepsilon _k(X) \overset{\operatorname {\scriptstyle def.}}{=}\operatorname {codim}_{V_k} \operatorname {Im}\tau _k. {\tag{2.9.3}} \end{equation}$$

Lemma 2.9.4.

(1) Take and fix a point $[B,i,j]\in {\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})$. Let $\tau _k$ be as in 2.9.1. If $[B, i, j]\in \pi ^{-1}({\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}}))$, then we have

$$\begin{equation} \operatorname {Im}\tau _k = V_k \qquad \text{for any $k\in I$}. {\tag{2.9.5}} \end{equation}$$

Moreover, the converse holds if we assume $\pi ([B,i,j])$ is regular in the sense of Definition 2.6.4. Namely under this assumption, $[B,i,j]\in \pi ^{-1}({\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}}))$ if and only if 2.9.5 holds.

(2) If ${\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}})\neq \emptyset$, then ${\mathbf{w}}-{\mathbf{v}}$ is dominant.

Proof.

(1) See 45, 4.7 for the first assertion. During the proof of 45, 7.2, we have shown the second assertion, using 45, 3.10 = Proposition 2.3.7.

(2) Consider the alternating sum of dimensions of the complex $C_k^\bullet$. It is equal to the alternating sum of dimensions of cohomology groups. It is nonnegative, if ${\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}})\neq \emptyset$ by Lemma 2.9.2 and (1). On the other hand, it is equal to

$$\begin{equation*} \sum _{h:\operatorname {out}(h)=k} \dim V_{\operatorname {in}(h)} + W_k - 2\dim V_k = \langle {\mathbf{w}}- {\mathbf{v}}, h_k\rangle . \end{equation*}$$

Thus we have the assertion.

■

3. Stratification of ${\mathfrak{M}}_0$

As was shown in 44, §6 and 45, 3.v, there exists a natural stratification of ${\mathfrak{M}}_0$ by conjugacy classes of stabilizers. A local topological structure of a neighborhood of a point in a stratum (e.g., the homology group of the fiber of $\pi$) was studied in 44, 6.10. We give a refinement in this section. We define a slice to a stratum, and study a local structure as a complex analytic space. Our technique is based on work of Sjamaar-Lerman 50 in the symplectic geometry and hence our transversal slice may not be algebraic. It is desirable to have a purely algebraic construction of a transversal slice, as Maffei did in a special case 42.

We fix dimension vectors ${\mathbf{v}}$, ${\mathbf{w}}$ and denote ${\mathbf{M}}({\mathbf{v}},{\mathbf{w}})$, ${\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})$ by ${\mathbf{M}}$, ${\mathfrak{M}}$ in this section.

3.1. Stratification

Definition 3.1.1 (cf. Sjamaar-Lerman 50).

For a subgroup $\widehat{G}$ of $G$ denote by ${\mathbf{M}}_{(\widehat{G})}$ the set of all points in ${\mathbf{M}}$ whose stabilizer is conjugate to $\widehat{G}$. A point $[(B,i,j)]\in {\mathfrak{M}}_0$ is said to be of $G$-orbit type $(\widehat{G})$ if its representative $(B,i,j)$ is in ${\mathbf{M}}_{(\widehat{G})}$. The set of all points of orbit type $(\widehat{G})$ is denoted by $({\mathfrak{M}}_0)_{(\widehat{G})}$.

The stratum $({\mathfrak{M}}_0)_{(1)}$ corresponding to the trivial subgroup $1$ is ${\mathfrak{M}_0^{\operatorname {reg}}}$ by definition. We have the following decomposition of ${\mathfrak{M}}_0$:

$$\begin{equation*} {\mathfrak{M}}_0 = \bigcup _{(\widehat{G})} ({\mathfrak{M}}_0)_{(\widehat{G})}, \end{equation*}$$

where the summation runs over the set of all conjugacy classes of subgroups of $G$.

For a more detailed description of $({\mathfrak{M}}_0)_{(\widehat{G})}$, see 44, 6.5, 45, 3.27.

3.2. Local normal form of the moment map

Let us recall the local normal form of the moment map following Sjamaar-Lerman 50.

Take $x\in {\mathfrak{M}}_0$ and fix its representative $m = (B,i,j)\in \mu ^{-1}(0)$. We suppose $m$ has a closed $G$-orbit and satisfies $\mu _{\mathbb{R}}(m) = 0$ by Proposition 2.4.1(1). Let $\widehat{G}$ be the stabilizer of $m$. It is the complexification of the stabilizer in $K = \prod \operatorname {U}(V_k)$ (see, e.g., 51, 1.6). Since $\mu (m) = 0$, the $G$-orbit $Gm = G/\widehat{G}$ through $m$ is an isotropic submanifold of ${\mathbf{M}}$. Let $\widehat{{\mathbf{M}}}$ be the quotient vector space $(T_m Gm)^\omega /T_m Gm$, where $T_m Gm$ is the tangent space of the orbit $Gm$, and $(T_m Gm)^\omega$ is the symplectic perpendicular of $T_m Gm$ in $T_m{\mathbf{M}}$, i.e., $\{ v\in T_m{\mathbf{M}}\mid \text{$\omega (v,w) = 0$ for all $w\in T_m Gm$}\}$. This is naturally a symplectic vector space. A vector bundle $T(Gm)^\omega /T(Gm)$ over $Gm$ is called the symplectic normal bundle. (In general, the symplectic normal bundle of an isotropic submanifold $S$ is defined by $TS^\omega /TS$.) It is isomorphic to $G\times _{\widehat{G}}\widehat{{\mathbf{M}}}$. (In 44, p.388, $\widehat{{\mathbf{M}}}$ was defined as the orthogonal complement of the quaternion vector subspace spanned by $T_m Km$ with respect to the Riemannian metric.) The action of $\widehat{G}$ on $\widehat{{\mathbf{M}}}$ preserves the induced symplectic structure on $\widehat{{\mathbf{M}}}$. Let $\widehat{\mu }\colon \widehat{{\mathbf{M}}}\to \widehat{{\mathfrak{g}}}^*$ be the corresponding moment map vanishing at the origin.

We choose an $\operatorname {Ad}(\widehat{G})$-invariant splitting ${\mathfrak{g}} = \widehat{{\mathfrak{g}}}\oplus \widehat{{\mathfrak{g}}}^\perp$ and its dual splitting ${\mathfrak{g}}^* = \widehat{{\mathfrak{g}}}^*\oplus \widehat{{\mathfrak{g}}}^{\perp *}$. Let us consider the natural action of $\widehat{G}$ on the product $T^*G\times \widehat{{\mathbf{M}}}= G\times {\mathfrak{g}}^*\times \widehat{{\mathbf{M}}}$. With the natural symplectic structure on $T^*G = G\times \mathfrak{g}^*$, we have the moment map

$$\begin{alignat*}{2} \widetilde{\mu }\colon & G\times {\mathfrak{g}}^*\times \widehat{{\mathbf{M}}}&\quad \to &\quad \widehat{{\mathfrak{g}}}^* \\ & (g,\xi ,\widehat{m}) & \mapsto & -\operatorname {pr}\xi + \widehat{\mu }(\widehat{m}), \end{alignat*}$$

where $\operatorname {pr}\xi$ is the projection of $\xi \in {\mathfrak{g}}^*$ to $\widehat{{\mathfrak{g}}}^*$. Zero is a regular value of $\widetilde{\mu }$, hence the symplectic quotient $\widetilde{\mu }^{-1}(0)/\widehat{G}$ is a symplectic manifold. It can be identified with $G\times _{\widehat{G}} \left( \widehat{{\mathfrak{g}}}^{\perp *}\times \widehat{{\mathbf{M}}}\right)$ via the map

$$\begin{equation*} G\times _{\widehat{G}} \left( \widehat{{\mathfrak{g}}}^{\perp *}\times \widehat{{\mathbf{M}}}\right) \ni \widehat{G}\cdot (g,\xi , \widehat{m}) \longmapsto \widehat{G}\cdot (g,\xi +\widehat{\mu }(\widehat{m}),\widehat{m}) \in \widetilde{\mu }^{-1}(0)/\widehat{G}. \end{equation*}$$

The embedding $G/\widehat{G}$ into $G\times _{\widehat{G}} \left( \widehat{{\mathfrak{g}}}^{\perp *}\times \widehat{{\mathbf{M}}}\right)$ is isotropic and its symplectic normal bundle is $G\times _{\widehat{G}}\widehat{{\mathbf{M}}}$. Thus two embeddings of $Gm\cong G/\widehat{G}$, one into ${\mathbf{M}}$ and the other into $G\times _{\widehat{G}} \left( \widehat{{\mathfrak{g}}}^{\perp *}\times \widehat{{\mathbf{M}}}\right)$, have the isomorphic symplectic normal bundles.

The $G$-equivariant version of Darboux-Moser-Weinstein’s isotropic embedding theorem (a special case of 50, 2.2) says the following:

Lemma 3.2.1.

A neighborhood of $Gm$ (in ${\mathbf{M}}$) is $G$-equivalently symplectomorphic to a neighborhood of $G/\widehat{G}$ embedded as the zero section of $G\times _{\widehat{G}} \left( \widehat{{\mathfrak{g}}}^{\perp *}\times \widehat{{\mathbf{M}}}\right)$ with the $G$-moment map given by the formula

$$\begin{equation*} \mu \left(\widehat{G}\cdot (g,\xi ,\widehat{m})\right) = \operatorname {Ad}^*(g)\left(\xi + \widehat{\mu }(\widehat{m})\right). \end{equation*}$$

(Here ‘symplectomorphic’ means that there exists a biholomorphism intertwining symplectic structures.)

Note that Sjamaar-Lerman worked on a real symplectic manifold with a compact Lie group action. Thus we need to take care when applying their result to our situation. Darboux-Moser-Weinstein’s theorem is based on the inverse function theorem, which we have both in the category of $C^\infty$-manifolds and in that of complex manifolds. A problem is that the domain of the symplectomorphism may not be chosen so that it covers the whole $Gm$ as it is noncompact. We can overcome this problem by taking a symplectomorphism defined in a neighborhood of the compact orbit $Km$ first, and then extending it to a neighborhood of $Gm$, as explained in the next three paragraphs. This approach is based on a result in 51.

A subset $A$ of a $G$-space $X$ is called orbitally convex with respect to the $G$-action if it is invariant under $K$ (= maximal compact subgroup of $G$) and for all $x\in A$ and all $\xi \in \mathfrak{k}$ we have that both $x$ and $\exp (\sqrt {-1}\xi )x$ in $A$ implies that $\exp (\sqrt {-1}t\xi )x\in A$ for all $t\in [0,1]$. By 51, 1.4, if $X$ and $Y$ are complex manifolds with $G$-actions, and if $A$ is an orbitally convex open subset of $X$ and $f\colon A\to Y$ is a $K$-equivariant holomorphic map, then $f$ can be uniquely extended to a $G$-equivariant holomorphic map.

Suppose that $X$ is a Kähler manifold with a (real) moment map $\mu _{\mathbb{R}}\colon X\to \mathfrak{k}^*$ and that $x\in X$ is a point such that $\mu _{\mathbb{R}}(x)$ is fixed under the coadjoint action of $K$. Then 51, Claim 1.13 says that the compact orbit $Kx$ possesses a basis of orbitally convex open neighborhoods.

In our situation, we have a Kähler metric (§2.4) and we have assumed $\mu _{\mathbb{R}}(m) = 0$. Thus $Km$ possesses a basis of orbitally convex open neighborhoods, and we have Lemma 3.2.1.

Now we want to study local structures of ${\mathfrak{M}}_0$, ${\mathfrak{M}}$ using Lemma 3.2.1. First the equation $\mu = 0$ implies $\xi = 0$, ${\widehat{\mu }}(\widehat{m}) = 0$. Thus ${\mathfrak{M}}_0$ and ${\mathfrak{M}}$ are locally isomorphic to ‘quotients’ of $G\times _{\widehat{G}}(\{0\}\times {\widehat{\mu }}^{-1}(0))$ by $G$, i.e., ‘quotients’ of ${\widehat{\mu }}^{-1}(0)$ by $\widehat{G}$. Here the ‘quotients’ are taken in the sense of the geometric invariant theory. Following Proposition 2.3.2(1), we say a point $\widehat{m}\in {\widehat{\mu }}^{-1}(0)$ is stable if the closure of $\widehat{G}\cdot (m,z)$ does not intersect with the zero section of ${\widehat{\mu }}^{-1}(0)\times {\mathbb{C}}$ for $z\neq 0$. Here we lift the $\widehat{G}$-action to the trivial line bundle ${\widehat{\mu }}^{-1}(0)\times {\mathbb{C}}$ by $\widehat{g}\cdot (\widehat{m},z) = (\widehat{g}\cdot \widehat{m}, \chi (g)^{-1}z)$, where $\chi$ is the restriction of the one-parameter subgroup used in §2.2. Let ${\widehat{\mu }}^{-1}(0)^{\operatorname {s}}$ be the set of stable points. As in §2.3, we have a morphism ${\widehat{\mu }}^{-1}(0)^{\operatorname {s}}/\widehat{G} \to {\widehat{\mu }}^{-1}(0)\mathord{\sslash }\widehat{G}$, which we denote by $\widehat{\pi }$. By 51, Proposition 2.7, we may assume that the neighborhood of $Gm$ in Lemma 3.2.1 is saturated, i.e., the closure of the $G$-orbit of a point in the neighborhood is contained in the neighborhood. Thus under the symplectomorphism in Lemma 3.2.1, (i) closed $G$-orbits are mapped to closed $\widehat{G}$-orbits, and (ii) the stability conditions are interchanged.

Proposition 3.2.2.

There exist a neighborhood $U$ (resp. $U^\perp$) of $x\in {\mathfrak{M}}_0$ (resp. $0\in {\widehat{\mu }}^{-1}(0)\mathord{\sslash }\widehat{G}$) and biholomorphic maps $\Phi \colon U\to U^\perp$, $\widetilde{\Phi }\colon \pi ^{-1}(U)\to {\widehat{\pi }}^{-1}(U^\perp )$ such that the following diagram commutes:

$$\begin{equation*} \begin{CD} \pi ^{-1}(U) @>\widetilde{\Phi }>\cong > {\widehat{\pi }}^{-1}(U^\perp ) \\@V{\pi }VV @VV\widehat{\pi }V \\U @>\Phi >\cong > U^\perp \end{CD} \end{equation*}$$

In particular, $\pi ^{-1}(x) = {\mathfrak{M}}_x$ is biholomorphic to ${\widehat{\pi }}^{-1}(0)$.

Furthermore, under $\Phi$, a stratum $({\mathfrak{M}}_0)_{(H)}$ of ${\mathfrak{M}}_0$ is mapped to a stratum $\left({\widehat{\mu }}^{-1}(0)\mathord{\sslash }\widehat{G}\right)_{(H)}$, which is defined as in Definition 3.1.1. (If $({\mathfrak{M}}_0)_{(H)}$ intersects with $U$, then $H$ is conjugate to a subgroup of $\widehat{G}$.)

The above discussion shows Proposition 3.2.2 except for the last assertion. The last assertion follows from the argument in 50, p.386.

3.3. Slice

By 44, p.391, we have a $\widehat{G}$-invariant splitting $\widehat{{\mathbf{M}}}= T\times T^\perp$, where $T$ is the tangent space $T_x({\mathfrak{M}}_0)_{(\widehat{G})}$ of the stratum containing $x$, and $\widehat{G}$ acts trivially on $T$. Thus we have

$$\begin{gather*} {\widehat{\mu }}^{-1}(0)\mathord{\sslash }\widehat{G} \cong T \times \left(T^\perp \cap \widehat{\mu }^{-1}(0)\right) \mathord{\sslash }\widehat{G}, \\ {\widehat{\mu }}^{-1}(0)^{\operatorname {s}}/\widehat{G} \cong T \times \left(T^\perp \cap \widehat{\mu }^{-1}(0)^{\operatorname {s}}\right)/ \widehat{G}. \end{gather*}$$

Furthermore, it was proved that $\left(T^\perp \cap \widehat{\mu }^{-1}(0)\right) \mathord{\sslash }\widehat{G}$ and $\left(T^\perp \cap \widehat{\mu }^{-1}(0)^{\operatorname {s}}\right)/ \widehat{G}$ are quiver varieties associated with a certain graph possibly different from the original one, and possibly with edge loops. Replacing $U^\perp$ if necessary, we may assume that $U^\perp$ is a product of a neighborhood $U_T$ of $0$ in $T$ and $U_{\mathfrak{S}}$ of $0$ in $\left(T^\perp \cap \widehat{\mu }^{-1}(0)\right) \mathord{\sslash }\widehat{G}.$ We define a transversal slice to $({\mathfrak{M}}_0)_{(\widehat{G})}$ at $x$ as

$$\begin{equation*} \mathfrak{S} \overset{\operatorname {\scriptstyle def.}}{=}\Phi ^{-1}\left(U^\perp \cap \left(\{ 0\} \times \left(T^\perp \cap \widehat{\mu }^{-1}(0)\right) \mathord{\sslash }\widehat{G} \right)\right) = \Phi ^{-1}\left(\{0\}\times U_{\mathfrak{S}}\right). \end{equation*}$$

Since $\Phi$ is a local biholomorphism, this slice $\mathfrak{S}$ satisfies the properties in 13, 3.2.19, i.e., there exists a biholomorphism

$$\begin{equation*} \left(U\cap ({\mathfrak{M}}_0)_{(\widehat{G})}\right)\times \mathfrak{S} \xrightarrow {\cong } U \end{equation*}$$

which induces biholomorphisms between factors

$$\begin{equation*} \{ x\} \times \mathfrak{S} \xrightarrow {\cong } \mathfrak{S}, \qquad \left(U\cap ({\mathfrak{M}}_0)_{(\widehat{G})}\right)\times \{x\} \xrightarrow {\cong } \left(U\cap ({\mathfrak{M}}_0)_{(\widehat{G})}\right). \end{equation*}$$

Remark 3.3.1.

Our construction gives a slice to a stratum in

$$\begin{equation*} {\mathfrak{M}}_{(0,\zeta _{\mathbb{C}})}= \mu ^{-1}(-\zeta _{\mathbb{C}})\mathord{\sslash }G_{\mathbf{v}} \end{equation*}$$

for general $\zeta _{\mathbb{C}}$. (See 44, p.371 and Theorem 3.1 for the definition of ${\mathfrak{M}}_{(0,\zeta _{\mathbb{C}})}$.) In particular, the fiber $\pi ^{-1}(x)$ of $\pi \colon {\mathfrak{M}}_{(\zeta _{\mathbb{R}},\zeta _{\mathbb{C}})}\to {\mathfrak{M}}_{(0,\zeta _{\mathbb{C}})}$ is isomorphic to the fiber of $\left( T^\perp \cap \widehat{\mu }^{-1}(0)^{\operatorname {s}}\right)/ \widehat{G} \to \left( T^\perp \cap \widehat{\mu }^{-1}(0)\right)\mathord{\sslash }\widehat{G}$ at $0$. This is a refinement of 44, 6.10, where an isomorphism between homology groups was obtained. We also remark that this gives a proof of smallness of

$$\begin{equation*} \pi \colon \bigsqcup _{\zeta _{\mathbb{C}}} {\mathfrak{M}}_{(\zeta _{\mathbb{R}},\zeta _{\mathbb{C}})} \to \bigsqcup _{\zeta _{\mathbb{C}}}{\mathfrak{M}}_{(0,\zeta _{\mathbb{C}})} \end{equation*}$$

which was observed by Lusztig when ${\mathfrak{g}}$ is of type $ADE$ 40. An essential point is, as remarked in 44, 6.11, that $\left( T^\perp \cap \widehat{\mu }^{-1}(0)^{\operatorname {s}}\right)/ \widehat{G}$ is diffeomorphic to an affine algebraic variety, and its homology group vanishes for degree greater than its complex dimension.

For our application, we only need the case when $x$ is regular, i.e., $x\in {\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}}^0,{\mathbf{w}})$ for some ${\mathbf{v}}^0$. Then, by 44, p.392, $\left(T^\perp \cap \widehat{\mu }^{-1}(0)\right) \mathord{\sslash }\widehat{G}$ and $\left(T^\perp \cap \widehat{\mu }^{-1}(0)\right)^{\operatorname {s}}/ \widehat{G}$ are isomorphic to the quiver varieties ${\mathfrak{M}}_0({\mathbf{v}}_s,{\mathbf{w}}_s)$ and ${\mathfrak{M}}({\mathbf{v}}_s,{\mathbf{w}}_s)$, associated with the original graph with dimension vector

$$\begin{equation*} {\mathbf{v}}_s = {\mathbf{v}}- {\mathbf{v}}^0, \qquad {\mathbf{w}}_s = {\mathbf{w}}- {\mathbf{C}}{\mathbf{v}}^0, \end{equation*}$$

where

$$\begin{equation*} {\mathbf{C}}{\mathbf{v}}^0 = \sum _{k\in I} \left(2 v^0_k - a_{kl}\, v^0_l\right)\Lambda _k \qquad \text{if }{\mathbf{v}}^0 = \sum _{k\in I} v^0_k \alpha _k \end{equation*}$$

in Convention 2.1.4.

Theorem 3.3.2.

Suppose that $x\in {\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}}^0,{\mathbf{w}})$ as above. Then there exist neighborhoods $U$, $U_T$, $U_{\mathfrak{S}}$ of $x\in {\mathfrak{M}}_0 = {\mathfrak{M}}_0({\mathbf{v}},{\mathbf{w}})$, $0\in T$, $0\in {\mathfrak{M}}_0({\mathbf{v}}_s,{\mathbf{w}}_s)$ respectively, and biholomorphic maps $U\to U_T\times U_{\mathfrak{S}}$, $\pi ^{-1}(U)\to U_T\times {\pi }^{-1}(U_{\mathfrak{S}})$ such that the following diagram commutes:

$$\begin{equation*} \begin{CD} {\mathfrak{M}}\;@. \supset \; @. \pi ^{-1}(U) @>>\cong > U_T \times \pi ^{-1}(U_{\mathfrak{S}}) \;@.\subset \;@. T\times {\mathfrak{M}}({\mathbf{v}}_s,{\mathbf{w}}_s) \\@. @. @V{\pi }VV @VV{\operatorname {id}\times \,\pi }V @.@. \\{\mathfrak{M}}_0\;@.\supset \; @. U @>>\cong > U_T\times U_{\mathfrak{S}} \;@.\subset \;@. T\times {\mathfrak{M}}_0({\mathbf{v}}_s,{\mathbf{w}}_s) \end{CD} \end{equation*}$$

In particular, $\pi ^{-1}(x) = {\mathfrak{M}}_x$ is biholomorphic to ${\mathfrak{L}}({\mathbf{v}}_s,{\mathbf{w}}_s)$.

Furthermore, a stratum of ${\mathfrak{M}}_0$ is mapped to a product of $U_T$ and a stratum of ${\mathfrak{M}}_0({\mathbf{v}}_s,{\mathbf{w}}_s)$.

Remark 3.3.3.

Suppose that $A$ is a subgroup of $G_{\mathbf{w}}\times {\mathbb{C}}^*$ fixing $x$. Since the $A$-action commutes with the $G$-action, $\widehat{{\mathbf{M}}}$ has an $A$-action. The above construction can be made $A$-equivariant. In particular, the diagram in Theorem 3.3.2 can be restricted to a diagram for $A$-fixed point sets.

4. Fixed point subvariety

Let $A$ be an abelian reductive subgroup of $G_{\mathbf{w}}\times {\mathbb{C}}^*$. In this section, we study the $A$-fixed point subvarieties ${\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})^A$ and ${\mathfrak{M}}_0({\mathbf{v}},{\mathbf{w}})^A$ of ${\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})$ and ${\mathfrak{M}}_0({\mathbf{v}},{\mathbf{w}})$.

4.1. A homomorphism attached to a component of ${\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})^A$

Suppose that $x\in {\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})^A$ is fixed by $A$. Take a representative $(B,i,j)\in \mu ^{-1}(0)^{\operatorname {s}}$ of $x$. For every $a\in A$, there exists $\rho (a)\in G_{\mathbf{v}}$ such that

$$\begin{equation} a \star (B,i,j) = \rho (a)^{-1} \cdot (B,i,j), {\tag{4.1.1}} \end{equation}$$

where the left hand side is the action defined in 2.7.2 and the right hand side is the action defined in 2.1.6. By the freeness of the $G_{\mathbf{v}}$-action on $\mu ^{-1}(0)^{\operatorname {s}}$ (see Proposition 2.3.2), $\rho (a)$ is uniquely determined by $a$. In particular, the map $a\mapsto \rho (a)$ is a homomorphism.

Let ${\mathfrak{M}}(\rho ) \subset {\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})^A$ be the set of fixed points $x$ such that 4.1.1 holds for some representative $(B,i,j)$ of $x$. Note that ${\mathfrak{M}}(\rho )$ depends only on the $G_{\mathbf{v}}$-conjugacy class of $\rho$. Since the $G_{\mathbf{v}}$-conjugacy class of $\rho$ is locally constant on ${\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})^A$, ${\mathfrak{M}}(\rho )$ is a union of connected components of ${\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})^A$. Later we show that ${\mathfrak{M}}(\rho )$ is connected under some assumptions (see Theorem 5.5.6). As in Proposition 2.3.8, we have

Proposition 4.1.2.

${\mathfrak{M}}(\rho )$ is homotopic to ${\mathfrak{M}}(\rho )\cap {\mathfrak{L}}({\mathbf{v}},{\mathbf{w}})$.

We regard $V$ as an $A$-module via $\rho$ and consider the weight space which corresponds to $\lambda \in \operatorname {Hom}(A,{\mathbb{C}}^*)$:

$$\begin{equation*} V(\lambda ) \overset{\operatorname {\scriptstyle def.}}{=}\{ v\in V\mid \rho (a)\cdot v = \lambda (a)v \}. \end{equation*}$$

We denote by $V_k(\lambda )$ the component of $V(\lambda )$ at the vertex $k$. We have $V = \bigoplus _\lambda V(\lambda )$. We regard $W$ as an $A$-module via the composition

$$\begin{equation*} A \hookrightarrow G_{\mathbf{w}}\times {\mathbb{C}}^* \xrightarrow {\text{projection}} G_{\mathbf{w}}. \end{equation*}$$

We also have the weight space decomposition $W = \bigoplus _\lambda W(\lambda )$, $W_k =\bigoplus _\lambda W_k(\lambda )$. We denote by $q$ the composition

$$\begin{equation*} A \hookrightarrow G_{\mathbf{w}}\times {\mathbb{C}}^* \xrightarrow {\text{projection}} {\mathbb{C}}^*. \end{equation*}$$

Then 4.1.1 is equivalent to

$$\begin{equation} \begin{split} B_{h}(V_{\operatorname {out}(h)}(\lambda )) \subset &\ V_{\operatorname {in}(h)}(q^{-m(h)-1}\lambda ), \\ i_k(W_k(\lambda )) \subset V_k(q^{-1}\lambda ),\ &\quad j_k(V_k(\lambda )) \subset W_k(q^{-1}\lambda ),\end{split} {\tag{4.1.3}} \end{equation}$$

where $m(h)$ is as in 2.7.1.

Lemma 4.1.4.

If $V_k(\lambda )\neq 0$, then $W_l(q^n\lambda )\neq 0$ for some $n$ and $l\in I$.

Proof.

Consider $\lambda$ satisfying $W_l(q^{n}\lambda ) = 0$ for any $l\in I$, $n\in {\mathbb{Z}}$. If we set

$$\begin{equation*} S_k \overset{\operatorname {\scriptstyle def.}}{=}\bigoplus _{\text{$\lambda $ as above}} V_k(\lambda ), \end{equation*}$$

then $S = (S_k)_{k\in I}$ is $B$-invariant and contained in $\operatorname {Ker}j$ by 4.1.3. Thus we have $S_k = 0$ by the stability condition.

■

The restriction of tautological bundles $V_k$, $W_k$ to ${\mathfrak{M}}(\rho )$ are bundles of $A$-modules. We have the weight decomposition $V_k = \bigoplus V_k(\lambda )$, $W_k = \bigoplus W_k(\lambda )$. We consider $V_k(\lambda )$, $W_k(\lambda )$ as vector bundles over ${\mathfrak{M}}(\rho )$.

Similarly, the restriction of the complex $C_k^\bullet$ in 2.9.1 decomposes as $C_k^\bullet = \bigoplus _\lambda C_{k,\lambda }^\bullet$, where

$$\begin{equation} C_{k,\lambda }^\bullet \equiv C_{k,\lambda }^\bullet (\rho ): V_k(q^2\lambda ) \xrightarrow {\sigma _{k,\lambda }} {\bigoplus _{h:\operatorname {in}(h)=k}} V_{\operatorname {out}(h)}(q^{m(h)+1}\lambda ) \oplus W_k(q\lambda ) \xrightarrow {\tau _{k,\lambda }} V_k(\lambda ). {\tag{4.1.5}} \end{equation}$$

Here $\sigma _{k,\lambda }$, $\tau _{k,\lambda }$ are restrictions of $\sigma _k$, $\tau _k$. When we want to emphasize that this is a complex over ${\mathfrak{M}}(\rho )$, we denote this by $C_{k,\lambda }^\bullet (\rho )$.

The tangent space of ${\mathfrak{M}}(\rho )$ at $[B,i,j]$ is the $A$-fixed part of the tangent space of ${\mathfrak{M}}$. Since the latter is the middle cohomology group of 2.1.8, the former is the middle cohomology group of the complex

$$\begin{equation*} \bigoplus _{\lambda ,k} \operatorname {End}\left(V_k(\lambda )\right) \longrightarrow \begin{matrix} \bigoplus _{\lambda , h} \operatorname {Hom}\left(V_{\operatorname {out}(h)}(\lambda ), V_{\operatorname {in}(h)}(q^{-m(h)-1}\lambda )\right) \\ \oplus \\ \bigoplus _{\lambda , k} \operatorname {Hom}\left(W_k(\lambda ), V_k(q^{-1}\lambda )\right) \\ \oplus \\ \bigoplus _{\lambda , k} \operatorname {Hom}\left(V_k(\lambda ), W_k(q^{-1}\lambda )\right) \end{matrix} \longrightarrow \bigoplus _{\lambda ,k} \operatorname {Hom}\left(V_k(\lambda ),V_k(q^{-2}\lambda )\right), \end{equation*}$$

where the differentials are the restrictions of $\iota$, $d\mu$ in 2.1.8. Those restrictions are injective and surjective respectively by Proposition 2.3.2. Hence we have the following dimension formula:

$$\begin{equation} \begin{split} & \dim {\mathfrak{M}}(\rho ) \\ &\qquad = \sum _{\lambda }\Big [ \sum _h \dim V_{\operatorname {out}(h)}(\lambda ) \dim V_{\operatorname {in}(h)}(q^{-m(h)-1}\lambda ) \\ & \quad \qquad \qquad + \sum _k \dim W_k(\lambda ) \left( \dim V_k(q^{-1}\lambda ) + \dim V_k(q\lambda ) \right)\\ & \quad \qquad \qquad \hphantom {+ \sum _k} \qquad - \dim V_k(\lambda )^2 - \dim V_k(\lambda )\dim V_k(q^{-2}\lambda )\Big ]. \end{split} {\tag{4.1.6}} \end{equation}$$

Recall that we have an isomorphism $\pi ^{-1}({\mathfrak{M}}_0^{\operatorname {reg}}) \cong {\mathfrak{M}}_0^{\operatorname {reg}}$ (Proposition 2.6.2). Let

$$\begin{equation} {\mathfrak{M}_0^{\operatorname {reg}}}(\rho ) \overset{\operatorname {\scriptstyle def.}}{=}\pi \left( \pi ^{-1}({\mathfrak{M}_0^{\operatorname {reg}}})\cap {\mathfrak{M}}(\rho ) \right) = \pi ^{-1}({\mathfrak{M}_0^{\operatorname {reg}}})\cap \pi \left({\mathfrak{M}}(\rho )\right). {\tag{4.1.7}} \end{equation}$$

By definition, $\pi ^{-1}({\mathfrak{M}_0^{\operatorname {reg}}}(\rho )) = \pi ^{-1}({\mathfrak{M}_0^{\operatorname {reg}}})\cap {\mathfrak{M}}(\rho )$ is an open subvariety of ${\mathfrak{M}}(\rho )$ which is isomorphic to ${\mathfrak{M}_0^{\operatorname {reg}}}(\rho )$ under $\pi$.

4.2. A sufficient condition for ${\mathfrak{M}}_0^A = \{ 0\}$

Let $a = (s,\varepsilon )$ be a semisimple element in $G_{\mathbf{w}}\times {\mathbb{C}}^*$ and let $A$ be the Zariski closure of $\{ a^n \mid n\in \mathbb{Z}\}$.

Definition 4.2.1.

We say $a$ is generic if ${\mathfrak{M}}_0({\mathbf{v}},{\mathbf{w}})^A = \{ 0\}$ for any ${\mathbf{v}}$. (This condition depends on ${\mathbf{w}}$.)

Proposition 4.2.2.

Assume that there is at most one edge joining two vertices of $I$, and that

$$\begin{equation*} \lambda /\lambda ' \notin \{ \varepsilon ^n \mid n\in \mathbb{Z}\setminus \{0\}\} \end{equation*}$$

for any pair of eigenvalues of $s\in G_{\mathbf{w}}$. (The condition for the special case $\lambda = \lambda '$ implies that $\varepsilon$ is not a root of unity.) Then $a = (s,\varepsilon )$ is generic.

Proof.

We prove ${\mathfrak{M}}_0({\mathbf{v}},{\mathbf{w}})^A = \{ 0\}$ by induction on ${\mathbf{v}}$. The assertion is trivial when ${\mathbf{v}}= 0$.

Take a point in ${\mathfrak{M}}_0({\mathbf{v}},{\mathbf{w}})^A$ and its representative $(B,i,j)$. As in 4.1.1, there exists $g\in G_{\mathbf{v}}$ such that

$$\begin{equation*} a \star (B,i,j) = g \cdot (B,i,j). \end{equation*}$$

We decompose $V$ into eigenspaces of $g$:

$$\begin{equation*} V = \bigoplus V(\lambda ), \qquad \text{where $V(\lambda ) \overset{\operatorname {\scriptstyle def.}}{=}\{ v\in V\mid g\cdot v = \lambda (a)v \}$}. \end{equation*}$$

We also decompose $W$ into eigenspaces of $s$ as $\bigoplus W(\lambda )$. Then 4.1.3 holds where $q$ is replaced by $\varepsilon$.

Choose and fix an eigenvalue $\mu$ of $s$. First suppose $V(\varepsilon ^n\mu )\neq 0$ for some $n$. Let

$$\begin{equation*} n_0 \overset{\operatorname {\scriptstyle def.}}{=}\max \left\{ n \mid V(\varepsilon ^n\mu )\neq 0 \right\}. \end{equation*}$$

Since $\varepsilon$ is not a root of unity, we have $\varepsilon ^n\mu \neq \varepsilon ^m\mu$ for $m\neq n$. Hence the above $n_0$ is well defined. By 4.1.3 (and $m(h) = 0$ from the assumption), we have $\operatorname {Im}B_h \cap V(\varepsilon ^{n_0}\mu ) = 0$. By the assumption, we have $W(\varepsilon ^{n_0+1}\mu ) = 0$, and hence $\operatorname {Im}i_k \cap V(\varepsilon ^{n_0}\mu ) = 0$ again by 4.1.3. Then we may assume the restriction of $(B,i,j)$ to $V(\varepsilon ^{n_0}\mu )$ is $0$ as in the proof of Lemma 2.5.3.

Thus the data $(B,i,j)$ is defined on the smaller subspace $V\ominus V(\varepsilon ^{n_0}\mu )$. Thus $(B,i,j) = 0$ by the induction hypothesis.

If $V(\varepsilon ^n\mu ) = 0$ for any $n$, we replace $\mu$. If we can find a $\mu '$ so that $V(\varepsilon ^n\mu ')\neq 0$ for some $n$, we are done. Otherwise, we have $V(\varepsilon ^n\mu ) = 0$ for any $n$, $\mu$, and we have $i = j = 0$ by 4.1.3. Then we choose $\mu$, which may not be an eigenvalue of $s$, so that $V(\mu ) \neq 0$ and repeat the above argument. (This is possible since we may assume $V\neq 0$.) We have $\operatorname {Im}B_h \cap V(\varepsilon ^{n_0}\mu ) = 0$ and the data $B$ is defined on the smaller subspace $V\ominus V(\varepsilon ^{n_0}\mu )$ as above.

■

5. Hecke correspondence and induction of quiver varieties

5.1. Hecke correspondence

Take dimension vectors ${\mathbf{w}}$, ${\mathbf{v}}^1$, ${\mathbf{v}}^2$ such that ${\mathbf{v}}^2 = {\mathbf{v}}^1 + \alpha _k$. Choose collections of vector spaces $W$, $V^1$, $V^2$, with $\dim W = {\mathbf{w}}$, $\dim V^a = {\mathbf{v}}^a$.

Let us consider the product ${\mathfrak{M}}({\mathbf{v}}^1,{\mathbf{w}})\times {\mathfrak{M}}({\mathbf{v}}^2,{\mathbf{w}})$. We denote by $V^1_k$ (resp. $V^2_k$) the vector bundle $V_k\boxtimes \mathcal{O}_{{\mathfrak{M}}({\mathbf{v}}^2,{\mathbf{w}})}$ (resp. $\mathcal{O}_{{\mathfrak{M}}({\mathbf{v}}^1,{\mathbf{w}})}\boxtimes V_k$). A point in ${\mathfrak{M}}({\mathbf{v}}^1,{\mathbf{w}})\times {\mathfrak{M}}({\mathbf{v}}^2,{\mathbf{w}})$ is denoted by $([B^1,i^1,j^1], [B^2, i^2, j^2])$. We regard $B^a$, $i^a$, $j^a$ ($a=1,2$) as homomorphisms between tautological bundles.

We define a three-term sequence of vector bundles over ${\mathfrak{M}}({\mathbf{v}}^1,{\mathbf{w}})\times {\mathfrak{M}}({\mathbf{v}}^2, {\mathbf{w}})$ by

$$\begin{equation} \operatorname {L}(V^1, V^2) \overset{\sigma }{\longrightarrow } q \operatorname {E}(V^1,V^2) \oplus q \operatorname {L}(W, V^2) \oplus q \operatorname {L}(V^1,W) \overset{\tau }{\longrightarrow } q^2 \operatorname {L}(V^1, V^2) \oplus q^2\mathcal{O}, {\tag{5.1.1}} \end{equation}$$

where

$$\begin{equation*} \begin{split} \sigma (\xi ) & = (B^2 \xi - \xi B^1) \oplus (-\xi i^1) \oplus j^2 \xi , \\ \tau (C\oplus a\oplus b) &= ( \varepsilon B^2 C + \varepsilon C B^1 + i^2 b + a j^1) \oplus \left( \operatorname {tr}(i^1 b) + \operatorname {tr}(a j^2)\right). \end{split} \end{equation*}$$

This is a complex, that is, $\tau \sigma = 0$, thanks to the equations $\varepsilon BB+ij = 0$ and $\operatorname {tr}(i^1 j^2 \xi ) = \operatorname {tr}(\xi i^1 j^2)$. Moreover, it is an equivariant complex with respect to the $G_{\mathbf{w}}\times {\mathbb{C}}^*$-action.

By 45, 5.2, $\sigma$ is injective and $\tau$ is surjective. Hence the quotient $\operatorname {Ker}\tau /\operatorname {Im}\sigma$ is a $G_{\mathbf{w}}\times {\mathbb{C}}^*$-equivariant vector bundle. Let us define an equivariant section $s$ of $\operatorname {Ker}\tau /\operatorname {Im}\sigma$ by

$$\begin{equation} s = \left( 0 \oplus (-i^2) \oplus j^1 \right) \bmod {\operatorname {Im}\sigma }, \tag{5.1.2} \end{equation}$$

where $\tau (s) = 0$ follows from $\varepsilon BB+ij=0$ and $\operatorname {tr}(\varepsilon B^1 B^1) = \operatorname {tr}(\varepsilon B^2 B^2) = 0$. The point $([B^1, i^1, j^1], [B^2, i^2, j^2])$ is contained in the zero locus $Z(s)$ of $s$ if and only if there exists $\xi \in \operatorname {L}(V^1, V^2)$ such that

$$\begin{equation} \xi B^1 = B^2 \xi , \quad \xi i^1 = i^2, \quad j^1 = j^2 \xi . {\tag{5.1.3}} \end{equation}$$

Moreover, $\operatorname {Ker}\xi$ is zero by the stability condition for $B^2$. Hence $\operatorname {Im}\xi$ is a subspace of $V^2$ with dimension ${\mathbf{v}}^1$ which is $B^2$-invariant and contains $\operatorname {Im}i^2$. Moreover, such $\xi$ is unique if we fix representatives $(B^1,i^1,j^1)$ and $(B^2,i^2,j^2)$. Hence we have an isomorphism between $Z(s)$ and the variety of all pairs $(B, i, j)$ and $S$ (modulo $G_{{\mathbf{v}}^2}$-action) such that

(a)

$(B, i, j)\in \mu ^{-1}(0)$ is stable, and

(b)

$S$ is a $B$-invariant subspace containing the image of $i$ with $\dim S = {\mathbf{v}}^1 = {\mathbf{v}}^2 - \alpha _k$.

Definition 5.1.4.

We call $Z(s)$ the Hecke correspondence, and denote it by ${\mathfrak{P}}_k({\mathbf{v}}^2,{\mathbf{w}})$. It is a $G_{\mathbf{w}}\times {\mathbb{C}}^*$-invariant closed subvariety.

Introducing a connection $\nabla$ on $\operatorname {Ker}\tau /\operatorname {Im}\sigma$, we consider the differential

$$\begin{equation*} \nabla s\colon T{\mathfrak{M}}({\mathbf{v}}^1,{\mathbf{w}})\oplus T{\mathfrak{M}}({\mathbf{v}}^2,{\mathbf{w}}) \to \operatorname {Ker}\tau / \operatorname {Im}\sigma \end{equation*}$$

of the section $s$. Its restriction to $Z(s) = {\mathfrak{P}}_k({\mathbf{v}}^2,{\mathbf{w}})$ is independent of the connection. By 45, 5.7, the differential $\nabla s$ is surjective over ${\mathfrak{P}}_k({\mathbf{v}}^2,{\mathbf{w}})$. Hence, ${\mathfrak{P}}_k({\mathbf{v}}^2,{\mathbf{w}})$ is nonsingular.

By the definition, the quotient $V_k^2/V_k^1$ defines a line bundle over ${\mathfrak{P}}_k({\mathbf{v}}^2,{\mathbf{w}})$.

5.2. Hecke correspondence and fixed point subvariety

Let $A$ be as in §4 and let ${\mathfrak{M}}(\rho )$ be as in §4.1 for $\rho \in \operatorname {Hom}(A,G_{\mathbf{v}})$.

Let us consider the intersection $({\mathfrak{M}}({\mathbf{w}})^A\times {\mathfrak{M}}({\mathbf{w}})^A)\cap {\mathfrak{P}}_k({\mathbf{v}}^2,{\mathbf{w}})$. It decomposes as

$$\begin{equation*} ({\mathfrak{M}}({\mathbf{w}})^A\times {\mathfrak{M}}({\mathbf{w}})^A)\cap {\mathfrak{P}}_k({\mathbf{v}}^2,{\mathbf{w}}) = \bigsqcup _{\rho ^1,\rho ^2} ({\mathfrak{M}}(\rho ^1)\times {\mathfrak{M}}(\rho ^2))\cap {\mathfrak{P}}_k({\mathbf{v}}^2,{\mathbf{w}}). \end{equation*}$$

Take a point $([B^1,i^1,j^1], [B^2,i^2,j^2])\in ({\mathfrak{M}}(\rho ^1)\times {\mathfrak{M}}(\rho ^2))\cap {\mathfrak{P}}_k({\mathbf{v}}^2,{\mathbf{w}})$. Then we have

$$\begin{equation*} a \star (B^p,i^p,j^p) = \rho ^p(a)^{-1} \cdot (B^p,i^p,j^p), \qquad a\in A\quad (p = 1,2), \end{equation*}$$

and there exists $\xi \in \operatorname {L}(V^1,V^2)$ such that

$$\begin{equation*} \xi B^1 = B^2 \xi , \quad \xi i^1 = i^2, \quad j^1 = j^2 \xi . \end{equation*}$$

By the uniqueness of $\xi$, we must have $\rho ^2(a)\xi = \xi \rho ^1(a)$, that is, $\xi \colon V^1\to V^2$ is $A$-equivariant. Since $\xi$ is injective, $V^1$ can be considered as an $A$-submodule of $V^2$.

If $V^1 = \bigoplus V^1(\lambda )$ and $V^2 = \bigoplus V^2(\lambda )$ are the weight decompositions, then there exists $\lambda _0$ such that

(a)

$V^1_l(\lambda ) \xrightarrow {\xi } V^2_l(\lambda )$ is an isomorphism if $\lambda \neq \lambda _0$ or $l\neq k$,

(b)

$V^1_k(\lambda _0) \xrightarrow {\xi } V^2_k(\lambda _0)$ is a codimension $1$ embedding.

5.3

We introduce a generalization of the Hecke correspondence. Let us define ${\mathfrak{P}}_k^{(n)}({\mathbf{v}},{\mathbf{w}})$ as

$$\begin{equation} {\mathfrak{P}}_k^{(n)}({\mathbf{v}},{\mathbf{w}}) \overset{\operatorname {\scriptstyle def.}}{=}\{ (B,i,j,S)\mid \text{$(B,i,j)\in {\mathbf{M}}({\mathbf{v}},{\mathbf{w}})$, $S\subset V$ as below}\} /G_{\mathbf{v}}, {\tag{5.3.1}} \end{equation}$$

(a)

$(B, i, j)\in \mu ^{-1}(0)^{\operatorname {s}}$,

(b)

$S$ is a $B$-invariant subspace containing the image of $i$ with $\dim S = {\mathbf{v}}- n\alpha _k$.

For $n = 1$, it is just ${\mathfrak{P}}_k({\mathbf{v}},{\mathbf{w}})$. We consider ${\mathfrak{P}}_k^{(n)}({\mathbf{v}},{\mathbf{w}})$ as a closed subvariety of ${\mathfrak{M}}({\mathbf{v}}-n\alpha _k,{\mathbf{w}})\times {\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})$ by setting

$$\begin{align*} & (B^1,i^1,j^1) \overset{\operatorname {\scriptstyle def.}}{=}\text{the restriction of $(B,i,j)$ to $S$},\\ & (B^2,i^2,j^2) \overset{\operatorname {\scriptstyle def.}}{=}(B,i,j). \end{align*}$$

We have a vector bundle of rank $n$ defined by $V_k^2/V_k^1$.

We shall show that ${\mathfrak{P}}_k^{(n)}({\mathbf{v}},{\mathbf{w}})$ is nonsingular later (see the proof of Lemma 11.2.3).

5.4. Induction

We recall some results in 45, §4. Let $Q_k({\mathbf{v}},{\mathbf{w}})$ be the middle cohomology of the complex 2.9.1, i.e.,

$$\begin{equation*} Q_k({\mathbf{v}},{\mathbf{w}}) \overset{\operatorname {\scriptstyle def.}}{=}\operatorname {Ker}\tau _k / \operatorname {Im}\sigma _k . \end{equation*}$$

We introduce the following subsets of ${\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})$ (cf. 34, 12.2):

$$\begin{equation} \begin{gathered} {\mathfrak{M}}_{k;n}({\mathbf{v}},{\mathbf{w}}) \overset{\operatorname {\scriptstyle def.}}{=}\left\{ [B,i,j]\in {\mathfrak{M}}({\mathbf{v}},{\mathbf{w}}) \Biggm | \operatorname {codim}_{V_k} \operatorname {Im}\tau _k = n \right\}, \\ {\mathfrak{M}}_{k;\le n}({\mathbf{v}},{\mathbf{w}}) \overset{\operatorname {\scriptstyle def.}}{=}\bigcup _{m\le n} {\mathfrak{M}}_{k;m}({\mathbf{v}},{\mathbf{w}}), \qquad {\mathfrak{M}}_{k;\ge n}({\mathbf{v}},{\mathbf{w}}) \overset{\operatorname {\scriptstyle def.}}{=}\bigcup _{m\ge n} {\mathfrak{M}}_{k;m}({\mathbf{v}},{\mathbf{w}}). \end{gathered} {\tag{5.4.1}} \end{equation}$$

Since ${\mathfrak{M}}_{k;\le n}({\mathbf{v}},{\mathbf{w}})$ is an open subset of ${\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})$, ${\mathfrak{M}}_{k;n}({\mathbf{v}},{\mathbf{w}})$ is a locally closed subvariety. The restriction of $Q_k({\mathbf{v}},{\mathbf{w}})$ to ${\mathfrak{M}}_{k;n}({\mathbf{v}},{\mathbf{w}})$ is a $G_{\mathbf{w}}\times {\mathbb{C}}^*$-equivariant vector bundle of rank $\langle h_k, {\mathbf{w}}- {\mathbf{v}}\rangle + n$, where we used Convention 2.1.4.

Replacing $V_k$ by $\operatorname {Im}\tau _k$, we have a natural map

$$\begin{equation} p\colon {\mathfrak{M}}_{k;n}({\mathbf{v}},{\mathbf{w}}) \to {\mathfrak{M}}_{k;0}({\mathbf{v}}- n\alpha _k,{\mathbf{w}}). {\tag{5.4.2}} \end{equation}$$

Note that the projection $\pi$ 2.3.4 factors through $p$. In particular, the fiber of $\pi$ is preserved under $p$.

Proposition 5.4.3.

Let $G(n,Q_k({\mathbf{v}}- n\alpha _k,{\mathbf{w}})|_{{\mathfrak{M}}_{k;0}({\mathbf{v}}-n\alpha _k,{\mathbf{w}})})$ be the Grassmann bundle of $n$-planes in the vector bundle obtained by restricting $Q_k({\mathbf{v}}- n\alpha _k,{\mathbf{w}})$ to ${\mathfrak{M}}_{k;0}({\mathbf{v}}-n\alpha _k,{\mathbf{w}})$. Then we have the following diagram:

$$\begin{equation*} \begin{CD} G(n,Q_k({\mathbf{v}}- n\alpha _k,{\mathbf{w}})|_{{\mathfrak{M}}_{k;0}({\mathbf{v}}-n\alpha _k,{\mathbf{w}})}) @>\pi >> {\mathfrak{M}}_{k;0}({\mathbf{v}}- n\alpha _k,{\mathbf{w}}) \\@VV{\cong }V @| \\{\mathfrak{M}}_{k;n}({\mathbf{v}},{\mathbf{w}}) @>p>> {\mathfrak{M}}_{k;0}({\mathbf{v}}- n\alpha _k,{\mathbf{w}}) \\@A{p_2}A{\cong }A @| \\{\mathfrak{P}}_k^{(n)}({\mathbf{v}},{\mathbf{w}}) \cap ({\mathfrak{M}}({\mathbf{v}}-n\alpha _k,{\mathbf{w}})\times {\mathfrak{M}}_{k;\le n}({\mathbf{v}},{\mathbf{w}})) @>{p_1}>> {\mathfrak{M}}_{k;0}({\mathbf{v}}-n\alpha _k,{\mathbf{w}}), \end{CD} \end{equation*}$$

where $\pi$ is the natural projection, and $p_1$ and $p_2$ are restrictions of the projections to the first and second factors. The kernel of the natural surjective homomorphism $p^* Q_k({\mathbf{v}}- n\alpha _k,{\mathbf{w}}) \to Q_k({\mathbf{v}},{\mathbf{w}})$ is isomorphic to the tautological vector bundle of the Grassmann bundle of the first row, and also to the the restriction of the vector bundle $V_k^2/V_k^1$ over ${\mathfrak{P}}_k^{(n)}({\mathbf{v}},{\mathbf{w}})$ in the third row.

Proof.

The proof is essentially contained in 45, 4.5. See also Proposition 5.5.2 for a similar result.

■

5.5. Induction for fixed point subvarieties

We consider an analogue of the results in the previous subsection for fixed point subvariety ${\mathfrak{M}}(\rho )$. Let us use notation as in §4.1, and suppose that $A$ is the Zariski closure of a semisimple element $a = (s,\varepsilon )\in G_{\mathbf{w}}\times {\mathbb{C}}^*$.

Let $Q_{k,\lambda }(\rho )$ be the middle cohomology of the complex $C_{k,\lambda }^\bullet (\rho )$ in 4.1.5, i.e.,

$$\begin{equation*} Q_{k,\lambda }(\rho ) \overset{\operatorname {\scriptstyle def.}}{=}\operatorname {Ker}\tau _{k,\lambda }/\operatorname {Im}\sigma _{k,\lambda }. \end{equation*}$$

Let

$$\begin{equation} {\mathfrak{M}}_{k;(n_\lambda )}(\rho ) \overset{\operatorname {\scriptstyle def.}}{=}\left\{ [B,i,j]\in {\mathfrak{M}}(\rho ) \Biggm | \text{$\operatorname {codim}_{V_k(\lambda )} \operatorname {Im}\tau _{k,\lambda } = n_\lambda $ for each $\lambda $}\right\}. {\tag{5.5.1}} \end{equation}$$

Replacing $V_k(\lambda )$ by $\operatorname {Im}\tau _{k,\lambda }$, we have a natural map

$$\begin{equation*} p^A \colon {\mathfrak{M}}_{k;(n_\lambda )}(\rho ) \to {\mathfrak{M}}_{k;(0)}(\rho '), \end{equation*}$$

where $\rho '\colon A\to G_{{\mathbf{v}}'}$ (${\mathbf{v}}' = {{\mathbf{v}}- \sum n_\lambda \alpha _k}$) is the homomorphism obtained from $\rho \colon A\to G_{\mathbf{v}}$ by replacing $V_k(\lambda )$ by its codimension $n_\lambda$ subspace. Its conjugacy class is independent of the choice of the subspace. This map is just the restriction of $p$ in the previous subsection.

For each $\lambda$, let $G(n_\lambda ,Q_{k,q^{-2}\lambda }(\rho ')|_{{\mathfrak{M}}_{k;(0)}(\rho ')})$ denote the Grassmann bundle of $n_\lambda$-planes in the vector bundle obtained by restricting $Q_{k,q^{-2}\lambda }(\rho ')$ to ${\mathfrak{M}}_{k;(0)}(\rho ')$. Let

$$\begin{equation*} \prod _\lambda G(n_\lambda ,Q_{k,q^{-2}\lambda }(\rho ')|_{{\mathfrak{M}}_{k;(0)}(\rho ')}) \end{equation*}$$

be their fiber product over ${\mathfrak{M}}_{k;(0)}(\rho ')$.

We have the following analogue of Proposition 5.4.3:

Proposition 5.5.2.

Suppose that $\varepsilon ^2 \neq 1$. We have the following diagram:

$$\begin{equation*} \begin{CD} {\displaystyle \prod _\lambda } G(n_\lambda ,Q_{k,q^{-2}\lambda }(\rho ')|_{{\mathfrak{M}}_{k;(0)}(\rho ')}) @>\pi >> {\mathfrak{M}}_{k;(0)}(\rho ') \\@VV{\cong }V @| \\{\mathfrak{M}}_{k;(n_\lambda )}(\rho ) @>p^A>> {\mathfrak{M}}_{k;(0)}(\rho '), \end{CD} \end{equation*}$$

where $\pi$ is the natural projection. For each $\lambda$, the kernel of the natural surjective homomorphism $(p^A)^* Q_{k,q^{-2}\lambda }(\rho ') \to Q_{k,q^{-2}\lambda }(\rho )$ is isomorphic to the tautological vector bundle of the Grassmann bundle. Moreover, we have

$$\begin{gather} n_\lambda \ge \max \left(0, -\operatorname {rank}C_{k,\lambda }^\bullet (\rho )\right), {\tag{5.5.3}}\\ \dim {\mathfrak{M}}_{k;(n_\lambda )}(\rho ) = \dim {\mathfrak{M}}(\rho ) - \sum _\lambda n_\lambda \left(\operatorname {rank}C_{k,\lambda }^\bullet (\rho ) + n_\lambda \right). {\tag{5.5.4}} \end{gather}$$

(Here $\operatorname {rank}$ of a complex means the alternating sum of dimensions of cohomology groups.)

Proof.

We have a surjective homomorphism $(p^A)^* Q_{k,q^{-2}\lambda }(\rho ') \to Q_{k,q^{-2}\lambda }(\rho )$ of codimension $n_\lambda$ over $M_{k;(n_\lambda )}(\rho )$. This gives a morphism from $M_{k;(n_\lambda )}(\rho )$ to the fiber product of Grassmann bundles. By a straightforward modification of the arguments in 45, 4.5, one can show that it is an isomorphism. The details are left to the reader. The assumption $\varepsilon ^2 \neq 1$ is used to distinguish $Q_{k,\lambda }$ and $Q_{k,q^{-2}\lambda }$.

Let us prove the remaining part 5.5.3, 5.5.4. First note that

$$\begin{equation*} \begin{split} \operatorname {rank}Q_{k,q^{-2}\lambda }(\rho ')|_{{\mathfrak{M}}_{k;(0)}(\rho ')} & = \operatorname {rank}C_{k,q^{-2}\lambda }^\bullet (\rho ') \\ & = \operatorname {rank}C_{k,q^{-2}\lambda }^\bullet (\rho ) + n_\lambda + n_{q^{-2}\lambda }. \end{split} \end{equation*}$$

Since we have an $n_\lambda$-dimensional subspace in $Q_{k,q^{-2}\lambda }(\rho ')|_{{\mathfrak{M}}_{k;(0)}(\rho ')}$, we must have

$$\begin{equation*} n_{q^{-2}\lambda } + \operatorname {rank}C_{k,q^{-2}\lambda }^\bullet \ge 0. \end{equation*}$$

Replacing $q^{-2}\lambda$ by $\lambda$, we get 5.5.3.

Moreover, we have

$$\begin{equation*} \dim {\mathfrak{M}}_{k;(n_\lambda )}(\rho ) = \dim {\mathfrak{M}}_{k;(0)}(\rho ') + \sum _\lambda n_\lambda \left(\operatorname {rank}C_{k,q^{-2}\lambda }^\bullet (\rho ) + n_{q^{-2}\lambda } \right). \end{equation*}$$

On the other hand, the dimension formula 4.1.6 implies

$$\begin{equation*} \dim {\mathfrak{M}}(\rho ) - \dim {\mathfrak{M}}(\rho ') = \sum _\lambda n_\lambda \left(\operatorname {rank}C_{k,\lambda }^\bullet (\rho ) + \operatorname {rank}C_{k,q^{-2}\lambda }^\bullet (\rho ) + n_{\lambda } + n_{q^{-2}\lambda }\right). \end{equation*}$$

Since ${\mathfrak{M}}_{k;(0)}(\rho ')$ is an open subset of ${\mathfrak{M}}(\rho ')$, we get 5.5.4.

■

Note that the inequality 5.5.3 implies that

$$\begin{equation*} n_\lambda \left(\operatorname {rank}C_{k,\lambda }^\bullet (\rho )+ n_\lambda \right) \ge 0 \end{equation*}$$

and the equality holds if and only if

$$\begin{equation*} n_\lambda = \max \left(0, -\operatorname {rank}C_{k,\lambda }^\bullet (\rho )\right). \end{equation*}$$

In particular, we have the following analog of 45, 4.6.

Corollary 5.5.5.

Suppose $\varepsilon ^2\neq 1$. On a nonempty open subset ${\mathfrak{M}}(\rho )$, we have

$$\begin{equation*} \operatorname {codim}_{V_k(\lambda )} \operatorname {Im}\tau _{k,\lambda } = \max \left(0, -\operatorname {rank}C_{k,\lambda }^\bullet (\rho )\right) \end{equation*}$$

for each $\lambda$. Also, the complement is a lower dimensional subvariety of ${\mathfrak{M}}(\rho )$.

As an application of this induction, we prove the following:

Theorem 5.5.6.

Assume that $\varepsilon$ is not a root of unity and there is at most one edge joining two vertices of $I$. Then ${\mathfrak{M}}(\rho )$ is connected if it is a nonempty set.

Proof.

We prove the assertion by induction on $\dim V$, $\dim W$. (The result is trivial when $V = W = 0$.)

We first make a reduction to the case when

$$\begin{equation} \operatorname {rank}C_{k,\lambda }^\bullet (\rho ) < 0 \quad \text{for some $k$, $\lambda $}. {\tag{5.5.7}} \end{equation}$$

Fix a $\mu \in \operatorname {Hom}(A,{\mathbb{C}}^*)$ and consider

$$\begin{equation*} n_0 \overset{\operatorname {\scriptstyle def.}}{=}\max \left\{ n \mid \text{$V_k(q^n\mu )\neq 0$ or $W_k(q^n\mu )\neq 0$ for some $k\in I$}\right\}. \end{equation*}$$

Since $\varepsilon$ is not a root of unity, we have $q^n\mu \neq q^m\mu$ for $m\neq n$. Hence the above $n_0$ is well defined.

Suppose $W_k(q^{n_0}\mu )\neq 0$. By 4.1.3 and the choice of $n_0$, we have

$$\begin{equation*} \operatorname {Im}j_k\cap W_k(q^{n_0}\mu ) = \{ 0\}. \end{equation*}$$

Let us replace $W_k(q^{n_0}\mu )$ by $0$. Namely we change $(\text{restriction of $i_k$})\colon W_k(q^{n_0}\mu )\to V_k(q^{n_0-1}\mu )$ to $0$ and all other data are unchanged. The equation $\mu (B,i,j) = 0$ and the stability condition are preserved by the replacement. Thus we have a morphism

$$\begin{equation*} {\mathfrak{M}}(\rho ) \to {\mathfrak{M}}'(\rho ), \end{equation*}$$

where ${\mathfrak{M}}'(\rho )$ is a fixed point subvariety of ${\mathfrak{M}}({\mathbf{v}},{\mathbf{w}}')$ obtained by the replacement. (This notation will not be used elsewhere. The data ${\mathbf{w}}$ is fixed elsewhere.)

Conversely, we can put any homomorphism $W_k(q^{n_0}\mu )\to V_k(q^{n_0-1}\mu )$ to get a point in ${\mathfrak{M}}(\rho )$ starting from a point in ${\mathfrak{M}}'(\rho )$. This shows that ${\mathfrak{M}}(\rho )$ is the total space of the vector bundle $\operatorname {Hom}(W_k(q^{n_0}\mu ), V_k(q^{n_0-1}\mu ))$ over ${\mathfrak{M}}'(\rho )$, where $W_k(q^{n_0}\mu )$ is considered as a trivial bundle. In particular, ${\mathfrak{M}}(\rho )$ is (nonempty and) connected if and only if ${\mathfrak{M}}(\rho ')$ is also. By the induction hypothesis, ${\mathfrak{M}}(\rho ')$ is connected and we are done.

Thus we may assume $V_k(q^{n_0}\mu )\neq 0$. Then $C_{k,q^{n_0}\mu }(\rho )$ consists of the last term by the choice of $n_0$. (Note $m(h) = 0$ under the assumption that there is at most one edge joining two vertices of $I$.) Hence we have 5.5.7 with $\lambda = q^{n_0}\mu$.

Now let us prove the connectedness of ${\mathfrak{M}}(\rho )$ under 5.5.7. By Corollary 5.5.5, we have

$$\begin{equation*} \dim {\mathfrak{M}}_{k;(n_\lambda )}(\rho ) < \dim {\mathfrak{M}}(\rho ) \end{equation*}$$

unless $n_\lambda = \max \left(0, -\operatorname {rank}C_{k,\lambda }^\bullet (\rho )\right)$ for each $\lambda$. Hence it is enough to prove the connectedness of ${\mathfrak{M}}_{k;(n_\lambda ^0)}(\rho )$ for $n_\lambda ^0 = \max \left(0, -\operatorname {rank}C_{k,\lambda }^\bullet (\rho )\right)$.

Let us consider the map $p^A\colon {\mathfrak{M}}_{k;(n_\lambda ^0)}(\rho )\to {\mathfrak{M}}_{k;(0)}(\rho ')$. By 5.5.7, $\dim V$ becomes smaller for ${\mathfrak{M}}_{k;(0)}(\rho ')$. Hence ${\mathfrak{M}}(\rho ')$ is connected by the induction hypothesis. Again by Corollary 5.5.5, ${\mathfrak{M}}_{k;(0)}(\rho ')$ is also connected. Since $p^A$ is a fiber product of Grassmann bundles, ${\mathfrak{M}}_{k;(n_\lambda ^0)}(\rho )$ is connected.

■

6. Equivariant $K$-theory

In this section, we review the equivariant $K$-theory of a quasi-projective variety with a group action. See 13, Chapter 5 for further details.

6.1. Definitions

Let $X$ be a quasi-projective variety over ${\mathbb{C}}$. Suppose that a linear algebraic group $G$ acts algebraically on $X$. Let $K^G(X)$ be the Grothendieck group of the abelian category of $G$-equivariant coherent sheaves on $X$. It is a module over $R(G)$, the representation ring of $G$.

A class in $K^G(X)$ represented by a $G$-equivariant sheaf $E$ will be denoted by $[E]$, or simply by $E$ if there is no fear of confusion.

The trivial line bundle of rank $1$, i.e., the structure sheaf, is denoted by $\mathcal{O}_X$. If the underlying space is clear, we simply write $\mathcal{O}$.

Let $K_G^0(X)$ be the Grothendieck group of the abelian category of $G$-equivariant algebraic vector bundles on $X$. This is also an $R(G)$-module. The tensor product $\otimes$ defines a structure of an $R(G)$-algebra on $K_G^0(X)$. Also, $K^G(X)$ has a structure of a $K_G^0(X)$-module by the tensor product:

$$\begin{equation} K_G^0(X)\times K^G(X)\ni ([E],[F]) \mapsto [E\otimes F] \in K^G(X). {\tag{6.1.1}} \end{equation}$$

Suppose that $Y$ is a $G$-invariant closed subvariety of $X$ and let $U = X\setminus Y$ be the complement. Two inclusions

$$\begin{equation*} Y \xrightarrow {i} X \xleftarrow {j} U \end{equation*}$$

induce an exact sequence

$$\begin{equation} \begin{CD} K^G(Y) @>{i_*}>> K^G(X) @>{j^*}>> K^G(U) @>>> 0, \end{CD} {\tag{6.1.2}} \end{equation}$$

where $i_*$ is given by $[E]\mapsto [i_* E]$ and $j^*$ is given by $[E]\mapsto [E|_U]$. (See 53.)

Suppose that $Y$ is a $G$-invariant closed subvariety of $X$ and that $X$ is nonsingular. Let $K^G(X;Y)$ be the Grothendieck group of the derived category of $G$-equivariant complexes $E^\bullet$ of algebraic vector bundles over $X$, which are exact outside $Y$ (see 4, §1). We have a natural homomorphism $K^G(X;Y)\to K^G(Y)$ by setting

$$\begin{equation*} [E^\bullet ] \mapsto \sum _i (-1)^i [\operatorname {gr}H^i(E^\bullet )]. \end{equation*}$$

Here $H^i(E^\bullet )$ is the $i$th cohomology sheaf of $E^\bullet$, which is a $G$-equivariant coherent sheaf on $X$ which is supported on $Y$. If $\mathfrak{I}_Y$ is the defining ideal of $Y$, we have $\mathfrak{I}_Y^N\cdot H^i(E^\bullet ) = 0$ for sufficiently large $N$. Then

$$\begin{equation*} \operatorname {gr}H^i(E^\bullet ) \overset{\operatorname {\scriptstyle def.}}{=}\bigoplus _j \mathfrak{I}_Y^j\cdot H^i(E^\bullet )/ \mathfrak{I}_Y^{j+1}\cdot H^i(E^\bullet ) \end{equation*}$$

is a sheaf on $Y$, and defines an element in $K^G(Y)$. Conversely if a $G$-equivariant coherent sheaf $F$ on $Y$ is given, we can take a resolution by a finite $G$-equivariant complex of algebraic vector bundles:

$$\begin{equation*} 0 \to E^{-n} \to E^{1-n} \to \dots \to E^0 \to i_* F \to 0, \end{equation*}$$

where $i\colon Y\to X$ denotes the inclusion. (See 13, 5.1.28.) This shows that the homomorphism $K^G(X;Y)\to K^G(Y)$ is an isomorphism. This relative $K$-group $K^G(X;Y)$ was not used in 13 explicitly, but many operations were defined by using it implicitly. When $Y = X$, $K^G(X;X)$ is isomorphic to $K_G^0(X)$. In particular, we have an isomorphism $K_G^0(X) \cong K^G(X)$ if $X$ is nonsingular.

We shall also use equivariant topological $K$-homology $K_{\operatorname {top}}^G(X)$. There are several approaches for the definition, but we take the one in 54, 5.3. There is a comparison map

$$\begin{equation*} K^G(X) \to K_{\operatorname {top}}^G(X) \end{equation*}$$

which satisfies obvious functorial properties.

Occasionally, we also consider the higher equivariant topological $K$-homology group $K_{1,\operatorname {top}}^G(X)$. (See 54, 5.3 again.) In this circumstance, $K_{\operatorname {top}}^G(X)$ may be written as $K_{0,\operatorname {top}}^G(X)$. But we do not use higher equivariant algebraic $K$-homology $K^G_i(X)$.

Suppose that $Y$ is a $G$-invariant closed subvariety of $X$ and let $U = X\setminus Y$ be the complement. Two inclusions

$$\begin{equation*} Y \xrightarrow {i} X \xleftarrow {j} U \end{equation*}$$

induce a natural exact hexagon

$$\begin{equation} \begin{CD} K_{0,\operatorname {top}}^G(Y) @>{i_*}>> K_{0,\operatorname {top}}^G(X) @>{j^*}>> K_{0,\operatorname {top}}^G(U) \\@AAA @. @VVV \\K_{1,\operatorname {top}}^G(U) @<{j^*}<< K_{1,\operatorname {top}}^G(X) @<{i_*}<< K_{1,\operatorname {top}}^G(Y), \end{CD} {\tag{6.1.3}} \end{equation}$$

for suitably defined $i_*$, $j^*$.

6.2. Operations on $K$-theory of vector bundles

If $E$ is a $G$-equivariant vector bundle, its rank and dual vector bundle will be denoted by $\operatorname {rank}E$ and $E^*$ respectively.

We extend $\operatorname {rank}$ and $*$ to operations on $K^0_G(X)$:

$$\begin{equation*} \operatorname {rank}\colon K^0_G(X) \to {\mathbb{Z}}^{\pi _0(X)}, \qquad {}^*\colon K^0_G(X) \to K^0_G(X), \end{equation*}$$

where $\pi _0(X)$ is the set of the connected components of $X$. Note that the rank of a vector bundle may not be a constant, when $X$ has several connected components. But we assume $X$ is connected in this subsection for simplicity. In general, operators below can be defined component-wisely.

If $L$ is a $G$-equivariant line bundle, we define $L^{\otimes r} = (L^*)^{\otimes (-r)}$ for $r < 0$. Thus we have $L^{\otimes r}\otimes L^{\otimes s} = L^{\otimes (r+s)}$ for $r,s\in {\mathbb{Z}}$.

If $E$ is a vector bundle, we define

$$\begin{equation*} \det E \overset{\operatorname {\scriptstyle def.}}{=}{\textstyle \bigwedge }^{\operatorname {rank}E} E, \qquad {\textstyle \bigwedge }_u E \overset{\operatorname {\scriptstyle def.}}{=}\sum _{i=0}^{\operatorname {rank}E} u^i {\textstyle \bigwedge }^i E. \end{equation*}$$

These operations can be extended to $K_G^0(X)$ of $G$-equivariant algebraic vector bundles:

$$\begin{equation*} \det \colon K_G^0(X)\to K_G^0(X), \quad {\textstyle \bigwedge }_u\colon K_G^0(X)\to [\mathcal{O}_X] + K_G^0(X)\otimes u{\mathbb{Z}}[[u]]. \end{equation*}$$

This is well defined since we have $\det F = \det E\otimes \det G$, ${\textstyle \bigwedge }_u F = {\textstyle \bigwedge }_u E \otimes {\textstyle \bigwedge }_u G$ for an exact sequence $0 \to E \to F \to G \to 0$.

Note the formula

$$\begin{equation*} {\textstyle \bigwedge }_u E = u^{\operatorname {rank}E} \det E\; {\textstyle \bigwedge }_{1/u} E^* \end{equation*}$$

for a vector bundle $E$. Using this formula, we expand ${\textstyle \bigwedge }_u E$ into the Laurent expansion also at $u=\infty$:

$$\begin{equation*} u^{-\operatorname {rank}E}(\det E)^* {\textstyle \bigwedge }_u E \in [\mathcal{O}_X] + K_G^0(X)\otimes u^{-1}{\mathbb{Z}}[[u^{-1}]]. \end{equation*}$$

6.3. Tor-product

(Cf. 4, 1.3, 13, 5.2.11.) Let $X$ be a nonsingular quasi-projective variety with a $G$-action. Let $Y_1$, $Y_2\subset X$ be $G$-invariant closed subvarieties of $X$. Suppose that $E_1^\bullet$ (resp. $E_2^\bullet$) is a $G$-equivariant complex of vector bundles over $X$ which is exact outside $Y_1$ (resp. $Y_2$). Then we can construct a complex

$$\begin{equation*} \cdots \longrightarrow \bigoplus _{p+q=k} E_1^p\otimes E_2^q \longrightarrow \bigoplus _{p+q=k+1} E_1^p\otimes E_2^q \longrightarrow \cdots \end{equation*}$$

with suitably defined differentials from the double complex $E_1^\bullet \otimes E_2^\bullet$. It is exact outside $Y_1\cap Y_2$. This construction defines an $R(G)$-bilinear pairing

$$\begin{equation*} K^G(X; Y_1) \times K^G(X; Y_2) \to K^G(X; Y_1\cap Y_2). \end{equation*}$$

Since we assume $X$ is nonsingular, we have $K^G(X;Y_1)\cong K^G(Y_1)$, $K^G(X;Y_2)\cong K^G(Y_2)$, $K^G(X;Y_1\cap Y_2)\cong K^G(Y_1\cap Y_2)$. Thus we also have an $R(G)$-bilinear pairing

$$\begin{equation*} K^G(Y_1) \times K^G(Y_2) \to K^G(Y_1\cap Y_2). \end{equation*}$$

We denote these operations by $\cdot \otimes ^L_X \cdot$. (It is denoted by $\otimes$ in 13.)

Lemma 6.3.1 (13, 5.4.10, 58, Lemma 1).

Let $Y_1$, $Y_2\subset X$ be nonsingular $G$-subvarieties with conormal bundles $T_{Y_1}^* X$, $T_{Y_2}^* X$. Suppose that $Y \overset{\operatorname {\scriptstyle def.}}{=}Y_1\cap Y_2$ is nonsingular and $TY_1|_Y \cap TZ_2|_Y = TY$, where $\ |_Y$ means the restriction to $Y$. Then for any $E_1\in K_G^0(Y_1)\cong K^G_0(Y_1)$, $E_2\in K_G^0(Y_2)\cong K^G_0(Y_2)$, we have

$$\begin{equation*} E_1 \otimes ^L_X E_2 = \sum _i (-1)^i {\textstyle \bigwedge }^i N \otimes E_1|_Y\otimes E_2|_Y \in K_G^0(Y)\cong K^G_0(Y), \end{equation*}$$

where $N \overset{\operatorname {\scriptstyle def.}}{=}T_{Y_1}^*X|_Y\cap T_{Y_2}^*X|_Y$.

6.4. Pull-back with support

(Cf. 4, 1.2, 13, 5.2.5.) Let $f\colon Y\to X$ be a $G$-equivariant morphism between nonsingular $G$-varieties. Suppose that $X'$ and $Y'$ are $G$-invariant closed subvarieties of $X$ and $Y$ respectively satisfying $f^{-1}(X')\subset Y'$. Then the pull-back

$$\begin{equation*} E^\bullet \mapsto f^* E^\bullet \end{equation*}$$

induces a homomorphism $K^G(X;X') \to K^G(Y;Y')$. Via isomorphisms $K^G(X')\cong K^G(X;X')$, $K^G(Y')\cong K^G(Y;Y')$, we get a homomorphism $K^G(X')\to K^G(Y')$. Note that this depends on the ambient spaces $X$, $Y$.

Let $f\colon Y\to X$ as above. Suppose that $X'_1, X'_2\subset X$, $Y'_1, Y'_2\subset Y$ are $G$-invariant closed subvarieties such that $f^{-1}(X'_a) \subset Y'_a$ for $a=1,2$. Then we have

$$\begin{equation} f^*(E_1\otimes ^L_X E_2) = f^*(E_1)\otimes ^L_Y f^*(E_2) {\tag{6.4.1}} \end{equation}$$

for $E_a\in K^G(X'_a)$ ($a=1,2$).

6.5. Push-forward

Let $f\colon X\to Y$ be a proper $G$-equivariant morphism between $G$-varieties (not necessarily nonsingular). Then we have a push-forward homomorphism $f_*\colon K^G(X)\to K^G(Y)$ defined by

$$\begin{equation*} f_*[E] \overset{\operatorname {\scriptstyle def.}}{=}\sum (-1)^i [R^i f_* E]. \end{equation*}$$

Suppose further that $X$ and $Y$ are nonsingular. If $X'\subset X$, $Y'\subset Y$ are $G$-invariant closed subvarieties, we have the following projection formula (13, 5.3.13):

$$\begin{equation} f_*(E \otimes ^L_X f^* F) = f_* E \otimes ^L_Y F \in K^G\left(f(X')\cap Y'\right) {\tag{6.5.1}} \end{equation}$$

for $E\in K^G(X')$, $F\in K^G(Y')$.

6.6. Chow group and homology group

Let $H_*(X,{\mathbb{Z}}) = \bigoplus _k H_k(X,{\mathbb{Z}})$ be the integral Borel-Moore homology of $X$. Let $A_*(X) = \bigoplus _k A_k(X)$ be the Chow group of $X$. We have a cycle map

$$\begin{equation*} A_*(X) \to H_*(X,{\mathbb{Z}}), \end{equation*}$$

which has certain functorial properties (see 19, Chapter 19).

If $Y$ is a closed subvariety of $X$ and $U = X\setminus Y$ is its complement, then we have exact sequences which are analogues of 6.1.2, 6.1.3:

$$\begin{gather} A_k(Y) \xrightarrow {i_*} A_k(X) \xrightarrow {j^*} A_k(U) \to 0, {\tag{6.6.1}}\\ \cdots \to H_k(Y,{\mathbb{Z}}) \xrightarrow {i_*} H_k(X,{\mathbb{Z}}) \xrightarrow {j^*} H_k(U,{\mathbb{Z}}) \xrightarrow {\partial _*} H_{k-1}(Y,{\mathbb{Z}}) \to \cdots . {\tag{6.6.2}} \end{gather}$$

We have operations on $A_*(X)$ and $H_*(X,{\mathbb{Z}})$ which are analogues of those in §6.3, §6.4, §6.5. (See 19.)

In the next section, we prove results for $K$-homology and the Chow group in parallel arguments. It is the reason why we avoid higher algebraic $K$-homology. There is no analogue for the Chow group.

7. Freeness

7.1. Properties $(S)$, $(T)$, $(T')$

Following 1439, we say that an algebraic variety $X$ has property $(S)$ if

(a)

$H_{\operatorname {odd}}(X, \mathbb{Z}) = 0$ and $H_{\operatorname {even}}(X,\mathbb{Z})$ is a free abelian group.

(b)

The cycle map $A_*(X)\to H_{\operatorname {even}}(X,\mathbb{Z})$ is an isomorphism.

Similarly, we say $X$ has property $(T)$ if

(a)

$K_{1,\operatorname {top}}(X) = 0$ and $K_{\operatorname {top}}(X) = K_{0,\operatorname {top}}(X)$ is a free abelian group.

(b)

The comparison map $K(X)\to K_{\operatorname {top}}(X)$ is an isomorphism.

Suppose that $X$ is a closed subvariety of a nonsingular variety $M$. We have a diagram (see 4)

$$\begin{equation*} \begin{CD} K(X)\otimes {\mathbb{Q}}@>\operatorname {ch}>> A_*(X)\otimes {\mathbb{Q}}\\@VVV @VVV \\K_{\operatorname {top}}(X)\otimes {\mathbb{Q}}@>\operatorname {ch}>> H_{\operatorname {even}}(X,{\mathbb{Q}}), \end{CD} \end{equation*}$$

where the horizontal arrows are local Chern character homomorphisms in algebraic and topological $K$-homologies respectively, the left vertical arrow is a comparison map, and the right vertical arrow is the cycle map. It is known that the upper horizontal arrow is an isomorphism (19, 15.2.16). Thus the composite $K(X)\otimes {\mathbb{Q}}\to H_{\operatorname {even}}(X,{\mathbb{Q}})$ is an isomorphism if $X$ has property $(S)$.

Assume that $X$ is nonsingular and projective. We define the bilinear pairing $K(X)\otimes K(X)\to \mathbb{Z}$ by

$$\begin{equation} F\otimes F' \mapsto p_*(F\otimes ^L_X F'), {\tag{7.1.1}} \end{equation}$$

where $p$ is the canonical map from $X$ to the point.

We say that $X$ has property $(T')$ if $X$ has property $(T)$ and the pairing 7.1.1 is perfect. (In 39, this property is called $(S')$.)

Let $G$ be a linear algebraic group. Let $X$ be an algebraic variety with a $G$-action. We say that $X$ has property $(T_G)$ if

(a)

$K^{G}_{1,\operatorname {top}}(X) = 0$ and $K_{\operatorname {top}}^{G}(X) = K^{G}_{0,\operatorname {top}}(X)$ is a free $R_G$-module.

(b)

The natural map $K^G(X)\to K_{\operatorname {top}}^{G}(X)$ is an isomorphism.

(c)

For a closed algebraic subgroup $H\subset G$, $H$-equivariant $K$-theories satisfy the above properties (a), (b), and the natural homomorphism $K^G(X)\otimes _{R(G)} R(H)\to K^H(X)$ is an isomorphism.

Suppose further that $X$ is smooth and projective. By the same formula as 7.1.1, we have a bilinear pairing $K^G(X)\otimes K^G(X)\to R(G)$. We say that $X$ has property $(T_G')$ if $X$ has property $(T_G)$ and this pairing is perfect.

A finite partition of a variety $X$ into locally closed subvarieties is said to be an $\alpha$-partition if the subvarieties in the partition can be indexed $X_1$, …, $X_n$ in such a way that $X_1\cup X_2\cup \cdots \cup X_i$ is closed in $X$ for $i=1$, …, $n$. The following is proved in 14, Lemma 1.8.

Lemma 7.1.2.

If $X$ has an $\alpha$-partition into pieces which have property $(S)$, then $X$ has property $(S)$.

The proof is based on exact sequences 6.6.1,6.6.2 in homology groups and Chow groups. Since we have corresponding exact sequences 6.1.2,6.1.3 in $K$-theory, we have the following.

Lemma 7.1.3.

Suppose that an algebraic variety $X$ has an action of a linear algebraic group $G$. If $X$ has an $\alpha$-partition into $G$-invariant locally closed subvarieties which have property $(T_G)$, then $X$ has property $(T_G)$.

Lemma 7.1.4.

Let $\pi \colon E\to X$ be a $G$-equivariant fiber bundle with affine spaces as fibers. Suppose that $\pi$ is locally a trivial $G$-equivariant vector bundle, i.e., a product of base and a vector space with a linear $G$-action. If $X$ has property $(T_G)$ (resp. $(S)$), then $E$ also has property $(T_G)$ (resp. $(S)$).

Proof.

We first show that $\pi ^*\colon K^G(X) \to K^G(E)$ is surjective. Choose a closed subvariety $Y$ of $X$ so that $E$ is a trivial $G$-bundle over $U = X\setminus Y$. There is a diagram

$$\begin{equation*} \begin{CD} K^G(Y) @>>> K^G(X) @>>> K^G(U) @>>> 0 \\@VV{\pi ^*}V @VV{\pi ^*}V @VV{\pi ^*}V \\K^G(\pi ^{-1}(Y)) @>>> K^G(E) @>>> K^G(\pi ^{-1}(U)) @>>> 0, \end{CD} \end{equation*}$$

with exact rows by 6.1.2. By a diagram chase it suffices to prove the surjectivity for the restrictions of $E$ to $U$ and to $Y$. By repeating the process on $Y$, it suffices to prove it for the case when $E$ is a trivial $G$-equivariant bundle. By Thom isomorphism 53, 4.1 $\pi ^*$ is an isomorphism if $E$ is a $G$-equivariant bundle. Thus we prove the assertion.

Let us repeat the same argument for $\pi ^*\colon K_{0,\operatorname {top}}^G(X)\to K_{0,\operatorname {top}}^G(E)$ and $\pi ^*: K_{1,\operatorname {top}}^G(X)\to K_{1,\operatorname {top}}^G(E)$ by replacing 6.1.2 by 6.1.3. By the five lemma both $\pi ^*$ are isomorphisms. In particular, we have $K_{1,\operatorname {top}}^G(E) \cong K_{1,\operatorname {top}}^G(X) = 0$ by assumption.

Consider the diagram

$$\begin{equation*} \begin{CD} K^G(X) @>\pi ^*>> K^G(E) \\@VVV @VVV \\K_{\operatorname {top}}^G(X) @>\pi ^*>> K_{\operatorname {top}}^G(E), \end{CD} \end{equation*}$$

where the vertical arrows are comparison maps. The left vertical arrow is an isomorphism by assumption. Thus the right vertical arrow is also an isomorphism by the commutativity of the diagram and what we just proved above. Condition (c) for $(T_G)$ can be checked in the same way, and $E$ has property $(T_G)$.

Property $(S)$ can be checked in the same way.

■

Lemma 7.1.5.

Let $X$ be a nonsingular quasi-projective variety with $G\times {\mathbb{C}}^*$-action with a Kähler metric $g$ such that

(a)

$g$ is complete,

(b)

$g$ is invariant under the maximal compact subgroup of $G\times {\mathbb{C}}^*$,

(c)

there exists a moment map $f$ associated with the Kähler metric $g$ and the $S^1$-action (the maximal compact subgroup of the second factor), and it is proper.

Let

$$\begin{equation*} L \overset{\operatorname {\scriptstyle def.}}{=}\{ x\in X \mid \text{$\displaystyle \lim _{t\to \infty } t\cdot x$ exists} \}. \end{equation*}$$

If the fixed point set $X^{{\mathbb{C}}^*}$ has property $(T_{G\times {\mathbb{C}}^*})$ (resp. $(S)$), then both $X$ and $L$ have property $(T_{G\times {\mathbb{C}}^*})$ (resp. $(S)$).

Furthermore, the bilinear pairing

$$\begin{equation} K^{G\times {\mathbb{C}}^*}(X) \times K^{G\times {\mathbb{C}}^*}(L) \ni (F, F') \longmapsto p_*(F\otimes ^L_X F')\in R(G\times {\mathbb{C}}^*) {\tag{7.1.6}} \end{equation}$$

is nondegenerate if $X^{{\mathbb{C}}^*}$ has property $(T_{G\times {\mathbb{C}}^*}')$. A similar intersection pairing between $A_*(X)$ and $A_*(L)$ is nondegenerate if $X^{{\mathbb{C}}^*}$ has property $(S)$. Here $p$ is the canonical map from $X$ to the point.

Proof.

By 2, 2.2 the moment map $f$ is a Bott-Morse function, and critical manifolds are the fixed point $X^{{\mathbb{C}}^*}$. Let $F_1$, $F_2$, …be the components of $X^{{\mathbb{C}}^*}$. By 2, §3, stable and unstable manifolds for the gradient flow of $-f$ coincide with $(\pm )$-attracting sets of Bialynicki-Birula decomposition 7:

$$\begin{equation*} S_k = \{ x\in X\mid \lim _{t\to 0} t\cdot x\in F_k \},\qquad U_k = \{ x\in X\mid \lim _{t\to \infty } t\cdot x \in F_k \}. \end{equation*}$$

These are invariant under the $G$-action since the $G$-action commutes with the ${\mathbb{C}}^*$-action.

Note that results in 2 are stated for compact manifolds, but the argument can be modified to our setting. A difference is that $\bigcup _k U_k = L$ is not $X$ unless $X$ is compact. On the other hand, $\bigcup _k S_k$ is $X$ since $f$ is proper.

As in 3, we can introduce an ordering on the index set $\{ k\}$ of components of $X^{{\mathbb{C}}^*}$ such that $X = \bigcup S_k$ is an $\alpha$-partition and $L = \bigcup U_k$ is an $\alpha$-partition with respect to the reversed order.

By 7 (see also 8 for analytic arguments), the maps

$$\begin{equation*} S_k \ni x \mapsto \lim _{t\to 0} t\cdot x\in F_k, \qquad U_k \ni x \mapsto \lim _{t\to \infty } t\cdot x\in F_k \end{equation*}$$

are fiber bundles with affine spaces as fibers. Furthermore, $S_k$ (resp. $U_k$) is locally isomorphic to a $G\times {\mathbb{C}}^*$-equivariant vector bundle by the proof. Thus $S_k$ and $U_k$ have properties $(S)$ and $(T_{G\times {\mathbb{C}}^*})$ by Lemma 7.1.4. Hence $X$ and $L$ have properties $(S)$ and $(T_{G\times {\mathbb{C}}^*})$ by Lemmas 7.1.2 and 7.1.3.

By the argument in 39, 1.7, 2.5, the pairing 7.1.6 can be identified with a pairing

$$\begin{equation*} \bigoplus _k K^{G\times {\mathbb{C}}^*} (F_k) \times \bigoplus _k K^{G\times {\mathbb{C}}^*} (F_k) \to R(G\times {\mathbb{C}}^*) \end{equation*}$$

of the form

$$\begin{equation*} \left( \sum _k \xi _k, \sum _k \xi '_k \right) = \sum _{k\ge k'} (\xi _k, \xi '_{k'})_{k,k'} \end{equation*}$$

for some pairing $(\ ,\ )_{k,k'}\colon K^{G\times {\mathbb{C}}^*}(F_k)\times K^{G\times {\mathbb{C}}^*}(F_k')\to R(G\times {\mathbb{C}}^*)$ such that $(\ ,\ )_{k,k}$ is the pairing 7.1.1 for $X = F_k$. Since $(\ ,\ )_{k,k}$ is nondegenerate for all $k$ by the assumption, 7.1.6 is also nondegenerate.

The proof of the statement for $A_*(X)$, $A_*(L)$ is similar. One uses the fact that the intersection pairing $A_*(F_k)\times A_*(F_k)\to {\mathbb{Z}}$ is nondegenerate under property $(S)$.

■

7.2. Decomposition of the diagonal

Proposition 7.2.1 (cf. 16, 13, 5.6.1).

Let $X$ be a nonsingular projective variety.

(1) Let $\mathcal{O}_{\Delta X}$ be the structure sheaf of the diagonal and $[\mathcal{O}_{\Delta X}]$ the corresponding element in $K(X\times X)$. Assume that

$$\begin{equation} [\mathcal{O}_{\Delta X}] = \sum _i \alpha _i\boxtimes \beta _i {\tag{7.2.2}} \end{equation}$$

holds for some $\alpha _i$, $\beta _i\in K(X)$. Then $X$ has property $(T')$.

(2) Let $G$ be a linear algebraic group. Suppose that $X$ has $G$-action and that 7.2.2 holds in $K^G(X\times X)$ for some $\alpha _i$, $\beta _i\in K^G(X)$. Then $X$ has property $(T'_G)$.

(3) Let $[\Delta X]$ be the class of the diagonal in $A(X\times X)$. Assume that

$$\begin{equation} [\Delta X] = \sum _i p_1^* a_i\cup p_2^*b_i {\tag{7.2.3}} \end{equation}$$

holds for some $a_i$, $b_i\in A(X)$. Then $X$ has property $(S)$.

Proof.

Let $p_a\colon X\times X\to X$ denote the projection to the $a$th factor ($a= 1,2$). Let $\Delta$ be the diagonal embedding $X \to X\times X$. Then we have $[\mathcal{O}_{\Delta X}] = \Delta _* [\mathcal{O}_X]$. Hence

$$\begin{alignat*}{2} p_{1*}\left(p_2^* F\otimes ^L_{X\times X} [\mathcal{O}_{\Delta X}]\right) & = p_{1*}\left(p_2^* F\otimes ^L_{X\times X} \Delta _* [\mathcal{O}_{X}]\right) &&\\ &= p_{1*}\Delta _* \left( \Delta ^* p_2^* F\otimes ^L_X [\mathcal{O}_X]\right) &\quad &\text{(by the projection formula)}\\ &= F\otimes ^L_X [\mathcal{O}_X] &&(p_1\circ \Delta = p_2\circ \Delta = \operatorname {id}_X)\\ &=F.&& \end{alignat*}$$

If we substitute 7.2.2 into the above, we get

$$\begin{equation} F = \sum _i p_{1*}\left(p_2^* F\otimes ^L_{X\times X} p_1^*\alpha _i \otimes ^L_{X\times X} p_2^*\beta _i\right) = \sum _i (F,\beta _i) \alpha _i. {\tag{7.2.4}} \end{equation}$$

In particular, $K(X)$ is spanned by $\alpha _i$’s.

If $mF = 0$ for some $m\in \mathbb{Z}\setminus \{0\}$, then $0 = (mF, \beta _i) = m(F,\beta _i)$. Hence we have $(F,\beta _i) = 0$. The above equality 7.2.4 implies $F = 0$. This means that $K(X)$ is torsion-free. Thus we could assume the $\alpha _i$’s are linearly independent in 7.2.2. Under this assumption, $\{\alpha _i\}$ is a basis of $K(X)$, and 7.2.4 implies that $\{ \beta _i \}$ is the dual basis.

If we perform the same computation in $K_{0,\operatorname {top}}(X)\oplus K_{1,\operatorname {top}}(X)$, we get the same result. In particular, $\{ \alpha _i \}$ is a basis of $K_{0,\operatorname {top}}(X)\oplus K_{1,\operatorname {top}}(X)$. However, $\alpha _i$, $\beta _i$ are in $K_{0,\operatorname {top}}(X)$, thus we have $K_{1,\operatorname {top}}(X)= 0$. We also have $K(X)\to K_{0,\operatorname {top}}(X)$ is an isomorphism. Thus $X$ has property $(T')$.

If $X$ has $G$-action and 7.2.2 holds in the equivariant $K$-group, we do the same calculation in the equivariant $K$-groups. Then the same argument shows that $X$ has property $(T'_G)$.

The assertion for Chow groups and homology groups can be proved in the same way.

■

7.3. Diagonal of the quiver variety

Let us recall the decomposition of the diagonal of the quiver variety defined in 45, Sect. 6. In this section, we fix dimension vectors ${\mathbf{v}}$, ${\mathbf{w}}$ and use the notation ${\mathfrak{M}}$ instead of ${\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})$.

Let us consider the product ${\mathfrak{M}}\times {\mathfrak{M}}$. We denote by $V^1_k$ (resp. $V^2_k$) the vector bundle $V_k\boxtimes \mathcal{O}_{\mathfrak{M}}$ (resp. $\mathcal{O}_{\mathfrak{M}}\boxtimes V_k$). A point in ${\mathfrak{M}}\times {\mathfrak{M}}$ is denoted by $([B^1,i^1,j^1], [B^2, i^2, j^2])$. We regard $B^a$, $i^a$, $j^a$ ($a=1,2$) as homomorphisms between tautological bundles.

We consider the following $G_{\mathbf{w}}\times {\mathbb{C}}^*$-equivariant complex of vector bundles over ${\mathfrak{M}}\times {\mathfrak{M}}$:

$$\begin{equation} \operatorname {L}(V^1, V^2) \overset{\sigma }{\longrightarrow } q\operatorname {E}(V^1,V^2) \oplus q\operatorname {L}(W, V^2) \oplus q\operatorname {L}(V^1,W) \overset{\tau }{\longrightarrow } q^2\operatorname {L}(V^1, V^2), {\tag{7.3.1}} \end{equation}$$

where

$$\begin{align*} \sigma (\xi ) & \overset{\operatorname {\scriptstyle def.}}{=}(B^2 \xi - \xi B^1) \oplus (-\xi i^1) \oplus j^2 \xi , \\ \tau (C\oplus a\oplus b) &\overset{\operatorname {\scriptstyle def.}}{=}( \varepsilon B^2 C + \varepsilon C B^1 + i^2 b + a j^1). \end{align*}$$

It was shown that $\sigma$ is injective and $\tau$ is surjective (cf. 45, 5.2). Thus $\operatorname {Ker}\tau /\operatorname {Im}\sigma$ is an equivariant vector bundle. We define an equivariant section $s$ of $\operatorname {Ker}\tau /\operatorname {Im}\sigma$ by

$$\begin{equation*} s \overset{\operatorname {\scriptstyle def.}}{=}\left( 0 \oplus (-i^2) \oplus j^1 \right) \bmod {\operatorname {Im}\sigma }. \end{equation*}$$

Then $([B^1,i^1,j^1], [B^2,i^2,j^2])$ is contained in the zero locus $Z(s)$ of $s$ if and only if there exists $\xi \in \operatorname {L}(V^1, V^2)$ such that

$$\begin{equation*} \xi B^1 = B^2 \xi , \quad \xi i^1 = i^2, \quad j^1 = j^2 \xi . \end{equation*}$$

Moreover $\xi$ is an isomorphism by the stability condition. Hence $Z(s)$ is equal to the diagonal $\Delta {\mathfrak{M}}$. If $\nabla$ is a connection on $\operatorname {Ker}\tau /\operatorname {Im}\sigma$, the differential $\nabla s\colon T({\mathfrak{M}}\times {\mathfrak{M}})\to \operatorname {Ker}\tau /\operatorname {Im}\sigma$ is surjective on $Z(s)=\Delta {\mathfrak{M}}$ (cf. 45, 5.7). In particular, we have an exact sequence

$$\begin{equation*} 0 \to {\textstyle \bigwedge }^{\max } (\operatorname {Ker}\tau /\operatorname {Im}\sigma )^* \to \cdots \to (\operatorname {Ker}\tau /\operatorname {Im}\sigma )^* \to \mathcal{O}_{{\mathfrak{M}}\times {\mathfrak{M}}} \to \mathcal{O}_{\Delta {\mathfrak{M}}} \to 0. \end{equation*}$$

In $K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}({\mathfrak{M}}\times {\mathfrak{M}})$, $\operatorname {Ker}\tau /\operatorname {Im}\sigma$ is equal to the alternating sum of terms of 7.3.1 which has a form $\sum \alpha _i \boxtimes \beta _i$ for some $\alpha _i,\beta _i\in K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}({\mathfrak{M}})$. Hence ${\mathfrak{M}}$ satisfies the conditions of Proposition 7.2.1 except the projectivity. Unfortunately, the projectivity is essential in the proof of Proposition 7.2.1. (We could not define $(p_1)_*$ otherwise.) Thus Proposition 7.2.1 is not directly applicable to ${\mathfrak{M}}$. In order to get rid of this difficulty, we consider the fixed point set ${\mathfrak{M}}^{{\mathbb{C}}^*}$ with respect to the ${\mathbb{C}}^*$-action.

For technical reasons, we need to use a ${\mathbb{C}}^*$-action, which is different from 2.7.2. Let ${\mathbb{C}}^*$ act on ${\mathbf{M}}$ by

$$\begin{equation} B_{h} \mapsto t B_{h}, \quad i \mapsto t i, \quad j \mapsto t j \qquad \text{for $t\in {\mathbb{C}}^*$}. {\tag{7.3.2}} \end{equation}$$

This induces a ${\mathbb{C}}^*$-action on ${\mathfrak{M}}$ and ${\mathfrak{M}}_0$ which commutes with the previous $G_{\mathbf{w}}\times {\mathbb{C}}^*$-action. (If the adjacency matrix satisfies ${\mathbf{A}}_{kl} \le 1$ for any $k, l\in I$, then the new ${\mathbb{C}}^*$-action coincides with the old one.) The tautological bundles $V_k$, $W_k$ become ${\mathbb{C}}^*$-equivariant vector bundles as before.

We consider the fixed point set ${\mathfrak{M}}^{{\mathbb{C}}^*}$. $[B,i,j]\in {\mathfrak{M}}$ is a fixed point if and only if there exists a homomorphism $\rho \colon {\mathbb{C}}^*\to G_{\mathbf{v}}$ such that

$$\begin{equation*} t\diamond (B,i,j) = \rho (t)^{-1}\cdot (B,i,j) \end{equation*}$$

as in §4.1. Here $\diamond$ denotes the new ${\mathbb{C}}^*$-action. We decompose the fixed point set ${\mathfrak{M}}^{{\mathbb{C}}^*}$ according to the conjugacy class of $\rho$:

$$\begin{equation*} {\mathfrak{M}}^{{\mathbb{C}}^*} = \bigsqcup {\mathfrak{M}}[\rho ]. \end{equation*}$$

Lemma 7.3.3.

${\mathfrak{M}}[\rho ]$ is a nonsingular projective variety.

Proof.

Since ${\mathfrak{M}}[\rho ]$ is a union of connected components (possibly single component) of the fixed point set of the ${\mathbb{C}}^*$-action on a nonsingular variety ${\mathfrak{M}}$, ${\mathfrak{M}}[\rho ]$ is nonsingular.

Suppose that $[B,i,j]\in {\mathfrak{M}}_0$ is a fixed point of the ${\mathbb{C}}^*$-action. It means that $(tB, ti, tj)$ lies in the closed orbit $G\cdot (B,i,j)$. But $(tB,ti,tj)$ converges to $0$ as $t\to 0$. Hence the closed orbit must be $\{0\}$. Since $\pi \colon {\mathfrak{M}}\to {\mathfrak{M}}_0$ is equivariant, ${\mathfrak{M}}^{{\mathbb{C}}^*}$ is contained in $\pi ^{-1}(0)$. In particular, ${\mathfrak{M}}[\rho ]$ is projective.

■

This lemma is not true for the original ${\mathbb{C}}^*$-action.

We restrict the complex 7.3.1 to ${\mathfrak{M}}[\rho ]\times {\mathfrak{M}}[\rho ]$. Then fibers of $V^1$ and $V^2$ become ${\mathbb{C}}^*$-modules and hence we can take the ${\mathbb{C}}^*$-fixed part of 7.3.1:

$$\begin{equation*} \operatorname {L}(V^1, V^2)^{{\mathbb{C}}^*} \overset{\sigma ^{{\mathbb{C}}^*}}{\longrightarrow } \left( q \operatorname {E}(V^1,V^2) \oplus q \operatorname {L}(W, V^2) \oplus q \operatorname {L}(V^1,W) \right)^{{\mathbb{C}}^*} \overset{\tau ^{{\mathbb{C}}^*}}{\longrightarrow } (q^2 \operatorname {L}(V^1, V^2))^{{\mathbb{C}}^*}, \end{equation*}$$

where $\sigma ^{{\mathbb{C}}^*}$ (resp. $\tau ^{{\mathbb{C}}^*}$) is the restriction of $\sigma$ (resp. $\tau$) to the ${{\mathbb{C}}^*}$-fixed part. Then $\sigma ^{{\mathbb{C}}^*}$ is injective and $\tau ^{{\mathbb{C}}^*}$ is surjective, and $\operatorname {Ker}\tau ^{{\mathbb{C}}^*}/\operatorname {Im}\sigma ^{{\mathbb{C}}^*}$ is a vector bundle which is the ${{\mathbb{C}}^*}$-fixed part of $\operatorname {Ker}\tau /\operatorname {Im}\sigma$.

The section $s$ takes values in $\operatorname {Ker}\tau ^{{\mathbb{C}}^*}/\operatorname {Im}\sigma ^{{\mathbb{C}}^*} = (\operatorname {Ker}\tau /\operatorname {Im}\sigma )^{{\mathbb{C}}^*}$. Considering it as a section of $\operatorname {Ker}\sigma ^{{\mathbb{C}}^*}/\operatorname {Im}\tau ^{{\mathbb{C}}^*}$, we denote it by $s^{{\mathbb{C}}^*}$. The zero locus $Z(s^{{\mathbb{C}}^*})$ is $Z(s)\cap ({\mathfrak{M}}[\rho ]\times {\mathfrak{M}}[\rho ])$ which is the diagonal $\Delta {\mathfrak{M}}[\rho ]$ of ${\mathfrak{M}}[\rho ]\times {\mathfrak{M}}[\rho ]$. Furthermore, the differential $\nabla s^{{\mathbb{C}}^*}\colon T({\mathfrak{M}}[\rho ]\times {\mathfrak{M}}[\rho ])\to (\operatorname {Ker}\tau /\operatorname {Im}\sigma )^{{\mathbb{C}}^*}$ is surjective on $Z(s^{{\mathbb{C}}^*})=\Delta {\mathfrak{M}}[\rho ]$.

Our original $G_{\mathbf{w}}\times {\mathbb{C}}^*$-action (defined in §2.7) commutes with the new ${\mathbb{C}}^*$-action. Thus ${\mathfrak{M}}[\rho ]$ has an induced $G_{\mathbf{w}}\times {\mathbb{C}}^*$-action. By the construction, $\operatorname {Ker}\tau ^{{\mathbb{C}}^*}/\operatorname {Im}\sigma ^{{\mathbb{C}}^*}$ is a $G_{\mathbf{w}}\times {\mathbb{C}}^*$-equivariant vector bundle, and $s^{{\mathbb{C}}^*}$ is an equivariant section.

Proposition 7.3.4.

${\mathfrak{M}}[\rho ]$ has properties $(S)$ and $(T_{G_{\mathbf{w}}\times {\mathbb{C}}^*}')$. Moreover, ${\mathfrak{M}}[\rho ]$ is connected.

Proof.

Let $\mathcal{O}_{\Delta {\mathfrak{M}}[\rho ]}$ be the structure sheaf of the diagonal considered as a sheaf on ${\mathfrak{M}}[\rho ]\times {\mathfrak{M}}[\rho ]$. By the above argument, the Koszul complex of $s^{{\mathbb{C}}^*}$ gives a resolution of $\mathcal{O}_{\Delta {\mathfrak{M}}[\rho ]}$:

$$\begin{multline*} 0 \to {\textstyle \bigwedge }^{\max } (\operatorname {Ker}\tau ^{{\mathbb{C}}^*}/\operatorname {Im}\sigma ^{{\mathbb{C}}^*})^* \to \cdots \to (\operatorname {Ker}\tau ^{{\mathbb{C}}^*}/\operatorname {Im}\sigma ^{{\mathbb{C}}^*})^*\\ \to \mathcal{O}_{{\mathfrak{M}}[\rho ]\times {\mathfrak{M}}[\rho ]} \to \mathcal{O}_{\Delta {\mathfrak{M}}[\rho ]} \to 0, \end{multline*}$$

where $\max = \operatorname {rank}\operatorname {Ker}\sigma ^{{\mathbb{C}}^*}/\operatorname {Im}\tau ^{{\mathbb{C}}^*}$. Thus we have the following equality in the Grothendieck group $K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}({\mathfrak{M}}[\rho ]\times {\mathfrak{M}}[\rho ])$:

$$\begin{equation*} [\mathcal{O}_{\Delta {\mathfrak{F}}}] = {\textstyle \bigwedge }_{-1} [(\operatorname {Ker}\tau ^{{\mathbb{C}}^*}/\operatorname {Im}\sigma ^{{\mathbb{C}}^*})^*]. \end{equation*}$$

Since $\sigma ^{{\mathbb{C}}^*}$ is injective and $\tau ^{{\mathbb{C}}^*}$ is surjective, we have

$$\begin{equation*} \begin{split} & [\operatorname {Ker}\tau ^{{\mathbb{C}}^*}/\operatorname {Im}\sigma ^{{\mathbb{C}}^*}] \\ &\qquad = - [ \operatorname {L}(V^1, V^2)^{{\mathbb{C}}^*} ] + \left[ (q\operatorname {E}(V^1,V^2) \oplus q \operatorname {L}(W, V^2) \oplus q\operatorname {L}(V^1,W))^{{\mathbb{C}}^*}\right]\\ &\qquad \quad - [(q^2 \operatorname {L}(V^1, V^2))^{{\mathbb{C}}^*}]. \end{split} \end{equation*}$$

Each factor of the right hand side can be written in the form $\sum _i \alpha _i\boxtimes \beta _i$ for some $\alpha _i, \beta _i\in K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}({\mathfrak{M}}[\rho ])$. For example, the first factor is equal to

$$\begin{equation*} \operatorname {L}(V^1, V^2)^{{\mathbb{C}}^*} = \bigoplus _m \operatorname {L}(V^1(m), V^2(m)), \end{equation*}$$

where $V^a(m)$ is the weight space of $V^a$, i.e.,

$$\begin{equation*} V^a(m) = \{ v\in V^a \mid t\diamond v = t^m v\}. \end{equation*}$$

The remaining factors have a similar description. Thus by Proposition 7.2.1, ${\mathfrak{M}}[\rho ]$ has property $(T_{G_{\mathbf{w}}\times {\mathbb{C}}^*}')$.

Moreover, the above shows that $K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}({\mathfrak{M}}[\rho ])$ is generated by exterior powers of $V_k(m)$, $W_k(m)$ and its duals (as an $R(G_{\mathbf{w}}\times {\mathbb{C}}^*)$-algebra). Note that these bundles have constant rank on ${\mathfrak{M}}[\rho ]$. If ${\mathfrak{M}}[\rho ]$ have components $M_1$, $M_2$, …, the structure sheaf of $M_1$ (extended to ${\mathfrak{M}}[\rho ]$ by setting $0$ outside) cannot be represented by $V_k(m)$, $W_k(m)$. This contradiction shows that ${\mathfrak{M}}[\rho ]$ is connected.

The assertion for Chow groups can be proved in exactly the same way. By the above argument, the fundamental class $[\Delta {\mathfrak{M}}[\rho ]]$ is the top Chern class of $\operatorname {Ker}\tau ^{{\mathbb{C}}^*}/\operatorname {Im}\sigma ^{{\mathbb{C}}^*}$, which can be represented as $\sum _i p_1^* a_i\cup p_2^* b_i$ for some $a_i$, $b_i\in A(X)$.

■

Theorem 7.3.5.

${\mathfrak{M}}$ and ${\mathfrak{L}}$ have properties $(S)$ and $(T_{G_{\mathbf{w}}\times {\mathbb{C}}^*})$. Moreover, the bilinear pairing

$$\begin{equation*} K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}({\mathfrak{M}}) \times K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}({\mathfrak{L}}) \ni (F, F') \longmapsto p_*(F\otimes ^L_{{\mathfrak{M}}} F')\in R(G_{\mathbf{w}}\times {\mathbb{C}}^*) \end{equation*}$$

is nondegenerate. A similar pairing between $A_*({\mathfrak{M}})$ and $A_*({\mathfrak{L}})$ is also nondegenerate. Here $p$ is the canonical map from ${\mathfrak{M}}$ to the point.

Proof.

We apply Lemma 7.1.5. By 44, 2.8, the metric on ${\mathfrak{M}}$ defined in §2.4 is complete. By the construction, it is invariant under $K_{\mathbf{w}}\times S^1$, where $K_{\mathbf{w}}= \prod \operatorname {U}(W_k)$ is the maximal compact subgroup of $G_{\mathbf{w}}$. (Note that the hyper-Kähler structure is not invariant under the $S^1$-action, but the metric is invariant.) The moment map for the $S^1$-actions is given by

$$\begin{equation*} \frac{1}{2} \left( \sum _h \| B_h \|^2 + \sum _k \left( \| i_k\|^2 + \| j_k \|^2 \right) \right). \end{equation*}$$

This is a proper function on ${\mathfrak{M}}$. Thus Lemma 7.1.5 is applicable. Note that we have ${\mathfrak{L}}= \{ x\in {\mathfrak{M}}\mid \text{$\lim _{t\to \infty } t\diamond x$ exists} \}$ as in 44, 5.8. (Though our ${\mathbb{C}}^*$-action is different from the one in 44, the same proof works.)

■

7.4. Fixed point subvariety

Let $A$ be an abelian reductive subgroup of $G_{\mathbf{w}}\times {\mathbb{C}}^*$ as in §4. Let ${\mathfrak{M}}^A$ and ${\mathfrak{L}}^A$ be the fixed point set in ${\mathfrak{M}}$ and ${\mathfrak{L}}$ respectively. Exactly as in the previous subsection, we have the following generalization of Theorem 7.3.5.

Theorem 7.4.1.

${\mathfrak{M}}^A$ and ${\mathfrak{L}}^A$ have properties $(T)$ and $(S)$. Moreover, the bilinear pairing

$$\begin{equation*} K({\mathfrak{M}}^A) \times K({\mathfrak{L}}^A) \ni (F, F') \longmapsto p_*(F\otimes ^L_{{\mathfrak{M}}^A} F')\in {\mathbb{Z}} \end{equation*}$$

is nondegenerate. A similar pairing between $A_*({\mathfrak{M}}^A)$ and $A_*({\mathfrak{L}}^A)$ is also nondegenerate. Here $p$ is the canonical map from ${\mathfrak{M}}^A$ to the point.

7.5. Connectedness of ${\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})$

Let us consider a natural homomorphism

$$\begin{equation} R(G_{\mathbf{w}}\times {\mathbb{C}}^*\times G_{\mathbf{v}}) \to K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}({\mathfrak{M}}), {\tag{7.5.1}} \end{equation}$$

which sends representations to bundles associated with tautological bundles. If we can apply Proposition 7.2.1 to ${\mathfrak{M}}$, then this homomorphism is surjective. Unfortunately we cannot apply Proposition 7.2.1 since ${\mathfrak{M}}$ is not projective. However, it seems reasonable to conjecture that the homomorphism 7.5.1 is surjective. In particular, it implies that ${\mathfrak{M}}$ is connected as in the proof of Proposition 7.3.4. This was stated in 45, 6.2. But the proof contains a gap since the function $\| s_1 \|$ may not be proper in general.

8. Convolution

Let $X_1$, $X_2$, $X_3$ be a nonsingular quasi-projective variety, and write $p_{ab}\colon X_1\times X_2\times X_3\to X_a\times X_b$ for the projection ($(a,b) = (1, 2), (2,3), (1,3)$).

Suppose $Z_{12}$ (resp. $Z_{23}$) is a closed subvariety of $X_1\times X_2$ (resp. $X_2\times X_3$) such that the restriction of the projection $p_{13}\colon p_{12}^{-1}(Z_{12})\cap p_{23}^{-1}(Z_{23})\to X_1\times X_3$ is proper. Let $Z_{12}\circ Z_{23} = p_{13}\left(p_{12}^{-1}(Z_{12})\cap p_{23}^{-1}(Z_{23})\right)$. We can define the convolution product $\ast \colon K(Z_{12})\otimes K(Z_{23})\to K(Z_{12}\circ Z_{23})$ by

$$\begin{equation*} K_{12}* K_{23} \overset{\operatorname {\scriptstyle def.}}{=}p_{13*} \left( p_{12}^* K_{12}\otimes ^L_{X_1\times X_2\times X_3} p_{23}^* K_{23}\right) \end{equation*}$$

for $K_{12}\in K(Z_{12}), K_{23}\in K(Z_{23})$.

Note that the convolution product depends on the ambient spaces $X_1$, $X_2$ and $X_3$. When we want to specify them, we say the convolution product relative to $X_1$, $X_2$, $X_3$.

In this section, we study what happens when $X_1$, $X_2$, $X_3$ are replaced by

(a)

submanifolds $S_1$, $S_2$, $S_3$ of $X_1$, $X_2$, $X_3$,

(b)

principal $G$-bundles $P_1$, $P_2$, $P_3$ over $X_1$, $X_2$, $X_3$.

Although we work on nonequivariant $K$-theory, the results extend to the case of equivariant $K$-theory, the Borel-Moore homology group, or any other reasonable theory in a straightforward way.

8.1

Before studying the above problem, we recall the following lemma which will be used several times.

Lemma 8.1.1.

In the above setting, we further assume that $X_1 = X_2$ and $Z_{12} = \operatorname {Image}\Delta _{X_1}$, where $\Delta _{X_1}$ is the diagonal embedding $X_1\to X_1\times X_2$. Then we have

$$\begin{equation*} (\Delta _{X_1})_*[E] * K_{23} = p_2^* [E] \otimes K_{23} \end{equation*}$$

for a vector bundle $E$ over $X_1$, where $p_2\colon X_2\times X_3\to X_2 = X_1$ is the projection, and $\otimes$ in the right hand side is the tensor product 6.1.1 between $K^0(Z_{23})$ and $K(Z_{23})$.

The proof is obvious from the definition, and is omitted.

8.2. Restriction of the convolution to submanifolds

Suppose we have nonsingular closed submanifolds $S_1$, $S_2$, $S_3$ of $X_1$, $X_2$, $X_3$ such that

$$\begin{equation} \left(S_1\times X_2\right) \cap Z_{12} \subset S_1\times S_2, \qquad \left(S_2\times X_3\right) \cap Z_{23} \subset S_2\times S_3. {\tag{8.2.1}} \end{equation}$$

By this assumption, we have

$$\begin{equation} (S_1\times X_3)\cap (Z_{12}\circ Z_{23}) \subset S_1\times S_3. {\tag{8.2.2}} \end{equation}$$

Let $Z_{12}'$ (resp. $Z_{23}'$) be the intersection $(S_1\times S_2)\cap Z_{12}$ (resp. $(S_2\times S_3)\cap Z_{23}$). By 8.2.2, we have $Z_{12}'\circ Z_{23}' = (S_1\times X_3)\cap (Z_{12}\circ Z_{23})$. We have the convolution product $\ast '\colon K(Z_{12}')\otimes K(Z_{23}') \to K(Z_{12}'\circ Z_{23}')$ relative to $S_1, S_2, S_3$:

$$\begin{equation*} K'_{12}*' K'_{23} \overset{\operatorname {\scriptstyle def.}}{=}p'_{13*} \left( p_{12}^{\prime *} K'_{12}\otimes ^L_{S_1\times S_2\times S_3} p_{23}^{\prime *} K'_{23}\right), \end{equation*}$$

where $p_{ab}'$ is the projection $S_1\times S_2\times S_3\to S_a\times S_b$.

We want to relate two convolution products $*$ and $*'$ via pull-back homomorphisms. For this purpose, we consider the inclusion $i_a\times \operatorname {id}_{X_b}\colon S_a\times X_b\to X_a\times X_b$, where $i_a$ is the inclusion $S_a\hookrightarrow X_a$ ($(a,b) = (1,2), (2,3), (1,3)$). By 8.2.1, we have a pull-back homomorphism

$$\begin{equation*} K(Z_{12})\cong K(X_1\times X_2; Z_{12}) \xrightarrow {(i_1\times \operatorname {id}_{X_2})^*} K(S_1\times X_2; Z_{12}\cap S_1\times X_2) \cong K(Z_{12}'). \end{equation*}$$

Similarly, we have

$$\begin{equation*} K(Z_{23})\xrightarrow {(i_2\times \operatorname {id}_{X_3})^*} K(Z_{23}'), \qquad K(Z_{12}\circ Z_{23})\xrightarrow {(i_1\times \operatorname {id}_{X_3})^*} K(Z_{12}'\circ Z_{23}'). \end{equation*}$$

Proposition 8.2.3.

For $K_{12}\in K(Z_{12}), K_{23}\in K(Z_{23})$, we have

$$\begin{equation} (i_1\times \operatorname {id}_{X_3})^*(K_{12} * K_{23}) = \left((i_1\times \operatorname {id}_{X_2})^*K_{12}\right) *' \left((i_2\times \operatorname {id}_{X_3})^* K_{23}\right). {\tag{8.2.4}} \end{equation}$$

Namely, the following diagram commutes:

$$\begin{equation*} \begin{CD} K(Z_{12})\otimes K(Z_{23}) @>{*}>> K(Z_{12}\circ Z_{23}) \\@V{(i_1\times \operatorname {id}_{X_2})^* \otimes (i_2\times \operatorname {id}_{X_3})^*}VV @VV{(i_1\times \operatorname {id}_{X_3})^*}V \\K(Z'_{12})\otimes K(Z'_{23}) @>{*'}>> K(Z'_{12}\circ Z'_{23}). \end{CD} \end{equation*}$$

Example 8.2.5.

Suppose $X_1 = X_2$, $S_1 = S_2$ and $Z_{12} = \operatorname {Image}\Delta _{X_1}$, where $\Delta _{X_1}$ is the diagonal embedding $X_1\to X_1\times X_2$. Then the above assumption $S_1\times X_2 \cap Z_{12} \subset S_1\times S_2$ is satisfied, and we have $Z_{12}' = \operatorname {Image}\Delta _{S_1}$, where $\Delta _{S_1}$ is the diagonal embedding $S_1\to S_1\times S_2$. If $E$ is a vector bundle over $X_1$, we have $(i_1\times \operatorname {id}_{X_2})^*(\Delta _{X_1})_*[E] = (\Delta _{S_1})_*[i_1^* E]$ by the base change 13, 5.3.15. By Lemma 8.1.1, we have

$$\begin{equation*} \begin{split} & (\Delta _{X_1})_*[E] * K_{23} = p_2^* [E] \otimes K_{23},\\ & (\Delta _{S_1})_*[i^* E] *' (i_2\times \operatorname {id}_{X_3})^* K_{23} = p_2^{\prime *}i_1^* [E] \otimes (i_2\times \operatorname {id}_{X_3})^* K_{23}, \end{split} \end{equation*}$$

where $p_2\colon X_2\times X_3\to X_2$, $p_2'\colon S_2\times X_3\to S_2$ are the projections. Note

$$\begin{equation*} \begin{split} p_2^{\prime *}i^* [E] \otimes (i_2\times \operatorname {id}_{X_3})^* K_{23} &= (i_2\times \operatorname {id}_{X_3})^* p_2^* [E] \otimes (i_2\times \operatorname {id}_{X_3})^* K_{23}\\ & = (i_2\times \operatorname {id}_{X_3})^*\left(p_2^*[E]\otimes K_{23}\right) \end{split} \end{equation*}$$

by 6.4.1. Hence we have 8.2.4 in this case.

Proof of Proposition 8.2.3.

In order to relate $*$ relative to $X_1$, $X_2$, $X_3$ and $*'$ relative to $S_1$, $S_2$, $S_3$, we replace $X_a$ by $S_a$ factor by factor.

Step 1. First we want to replace $X_1$ by $S_1$. We consider the following fiber square:

$$\begin{equation*} \begin{CD} S_1\times X_2\times X_3 @>{i_1\times \operatorname {id}_{X_2}\times \operatorname {id}_{X_3}}>> X_1\times X_2\times X_3 \\@V{p_{13}''}VV @VV{p_{13}}V \\S_1\times X_3 @>>{i_1\times \operatorname {id}_{X_3}}> X_1\times X_3, \end{CD} \end{equation*}$$

where $p_{13}''$ is the projection. We have

$$\begin{equation} \begin{split} & (i_1\times \operatorname {id}_{X_3})^* (K_{12} * K_{23}) = (i_1\times \operatorname {id}_{X_3})^* p_{13*} \left( p_{12}^* K_{12}\otimes ^L_{X_1\times X_2\times X_3} p_{23}^* K_{23}\right) \\ =\; & p_{13*}''(i_1\times \operatorname {id}_{X_2}\times \operatorname {id}_{X_3})^* \left(p_{12}^* K_{12}\otimes ^L_{X_1\times X_2\times X_3} p_{23}^* K_{23}\right) \\ =\; & p_{13*}'' \left( (i_1\times \operatorname {id}_{X_2}\times \operatorname {id}_{X_3})^* p_{12}^* K_{12} \otimes ^L_{S_1\times X_2\times X_3} (i_1\times \operatorname {id}_{X_2}\times \operatorname {id}_{X_3})^* p_{23}^* K_{23}\right), \end{split} {\tag{8.2.6}} \end{equation}$$

where we have used the base change (13, 5.3.15) in the second equality and 6.4.1 in the third equality. If $p_{12}''\colon S_1\times X_2\times X_3\to S_1\times X_2$ denotes the projection, we have $p_{12}\circ (i_1\times \operatorname {id}_{X_2}\times \operatorname {id}_{X_3}) = (i_1\times \operatorname {id}_{X_2})\circ p_{12}''$. Hence we get

$$\begin{equation*} (i_1\times \operatorname {id}_{X_2}\times \operatorname {id}_{X_3})^* p_{12}^* K_{12} = p_{12}^{\prime \prime *}(i_1\times \operatorname {id}_{X_2})^* K_{12}. \end{equation*}$$

Similarly, we have

$$\begin{equation*} (i_1\times \operatorname {id}_{X_2}\times \operatorname {id}_{X_3})^* p_{23}^* K_{23} = p_{23}^{\prime \prime *} K_{23}, \end{equation*}$$

where $p_{23}''\colon S_1\times X_2\times X_3\to X_2\times X_3$ is the projection. Substituting this into 8.2.6, we obtain

$$\begin{equation} (i_1\times \operatorname {id}_{X_3})^* (K_{12} * K_{23}) = p_{13*}'' \left( p_{12}^{\prime \prime *}(i_1\times \operatorname {id}_{X_2})^* K_{12} \otimes ^L_{S_1\times X_2\times X_3} p_{23}^{\prime \prime *} K_{23}\right). {\tag{8.2.7}} \end{equation}$$

Step 2. Next we replace $X_2$ by $S_2$. By 8.2.1, we have a homomorphism

$$\begin{equation*} (\operatorname {id}_{S_1}\times i_2)_*\colon K(Z_{12}')\cong K(S_1\times S_2; Z_{12}')\to K(S_1\times X_2; Z_{12}') \cong K(Z_{12}'), \end{equation*}$$

which is just the identity operator. We will consider $(i_1\times \operatorname {id}_{X_2})^* K_{12}\in K(Z_{12}')$ as an element of $K(S_1\times S_2; Z_{12}')$ or $K(S_1\times X_2; Z_{12}')$ interchangeably. We consider the fiber square

$$\begin{equation*} \begin{CD} S_1\times S_2 \times X_3 @>{p_{12}'''}>> S_1\times S_2 \\@V{\operatorname {id}_{S_1}\times i_2\times \operatorname {id}_{X_3}}VV @VV{\operatorname {id}_{S_1}\times i_2}V \\S_1\times X_2\times X_3 @>>{p_{12}''}> S_1\times X_2, \end{CD} \end{equation*}$$

where $p_{12}'''$ is the projection. By base change 13, 5.3.15, we get

$$\begin{equation*} \begin{split} p_{12}^{\prime \prime *}(i_1\times \operatorname {id}_{X_2})^* K_{12}& = p_{12}^{\prime \prime *}(\operatorname {id}_{S_1}\times i_2)_* (i_1\times \operatorname {id}_{X_2})^* K_{12}\\ & = (\operatorname {id}_{S_1}\times i_2\times \operatorname {id}_{X_3})_* p_{12}^{\prime \prime \prime *} (i_1\times \operatorname {id}_{X_2})^* K_{12}. \end{split} \end{equation*}$$

By the projection formula 6.5.1, we get

$$\begin{align*} & p_{12}^{\prime \prime *}(i_1\times \operatorname {id}_{X_2})^* K_{12} \otimes ^L_{S_1\times X_2\times X_3} p_{23}^{\prime \prime *} K_{23} \\ &= (\operatorname {id}_{S_1}\times i_2\times \operatorname {id}_{X_3})_* \left( p_{12}^{\prime \prime \prime *} (i_1\times \operatorname {id}_{X_2})^* K_{12} \otimes ^L_{S_1\times S_2\times X_3} (\operatorname {id}_{S_1}\times i_2\times \operatorname {id}_{X_3})^* p_{23}^{\prime \prime *}K_{23} \right). \end{align*}$$

Substituting this into 8.2.7, we have

$$\begin{equation} \begin{split} & (i_1\times \operatorname {id}_{X_3})^* (K_{12} * K_{23})\\ &\qquad = p_{13*}''(\operatorname {id}_{S_1}\times i_2\times \operatorname {id}_{X_3})_*\\ &\qquad \quad \times \left( p_{12}^{\prime \prime \prime *} (i_1\times \operatorname {id}_{X_2})^* K_{12} \otimes ^L_{S_1\times S_2\times X_3} (\operatorname {id}_{S_1}\times i_2\times \operatorname {id}_{X_3})^* p_{23}^{\prime \prime *}K_{23} \right) \\ &\qquad = p_{13*}'''\left( p_{12}^{\prime \prime \prime *} (i_1\times \operatorname {id}_{X_2})^* K_{12} \otimes ^L_{S_1\times S_2\times X_3} p_{23}^{\prime \prime \prime *}(i_2\times \operatorname {id}_{X_3})^* K_{23} \right), \end{split} {\tag{8.2.8}} \end{equation}$$

where $p_{13}'''\colon S_1\times S_2\times X_3 \to S_1\times X_3$, $p_{23}'''\colon S_1\times S_2\times X_3 \to S_2\times X_3$ are the projections, and we have used $p_{13}''' = p_{13}''\circ (\operatorname {id}_{S_1}\times i_2\times \operatorname {id}_{X_3})$ and $p_{23}''\circ (\operatorname {id}_{S_1}\times i_2\times \operatorname {id}_{X_3}) = (i_2\times \operatorname {id}_{X_3})\circ p_{23}'''$.

Step 3. We finally replace $X_3$ by $S_3$. By 8.2.1, we have a homomorphism

$$\begin{equation*} (\operatorname {id}_{S_2}\times i_3)_*\colon K(Z_{23}')\cong K(S_2\times S_3; Z_{23}')\to K(S_2\times X_3; Z_{23}') \cong K(Z_{23}'), \end{equation*}$$

which is just the identity operator. We consider the fiber square

$$\begin{equation*} \begin{CD} S_1\times S_2 \times S_3 @>{p_{23}'}>> S_2\times S_3 \\@V{\operatorname {id}_{S_1}\times \operatorname {id}_{S_2}\times i_3}VV @VV{\operatorname {id}_{S_2}\times i_3}V \\S_1\times S_2\times X_3 @>>{p_{23}'''}> S_2\times X_3. \end{CD} \end{equation*}$$

By base change 13, 5.3.15, we get

$$\begin{align*} p_{23}^{\prime \prime \prime *}(i_2\times \operatorname {id}_{X_3})^* K_{23} & = p_{23}^{\prime \prime \prime *}(\operatorname {id}_{S_2}\times i_3)_* (i_2\times \operatorname {id}_{X_3})^* K_{23}\\ & = (\operatorname {id}_{S_1}\times \operatorname {id}_{S_2}\times i_3)_* p_{23}^{\prime *} (i_2\times \operatorname {id}_{X_3})^* K_{23}. \end{align*}$$

Substituting this into 8.2.8, we obtain

$$\begin{align*} & (i_1\times \operatorname {id}_{X_3})^* (K_{12} * K_{23})\\ &\quad = p_{13*}''' \left(p_{12}^{\prime \prime \prime *} (i_1\times \operatorname {id}_{X_2})^* K_{12} \otimes ^L_{S_1\times S_2\times X_3} (\operatorname {id}_{S_1}\times \operatorname {id}_{S_2}\times i_3)_* p_{23}^{\prime *} (i_2\times \operatorname {id}_{X_3})^* K_{23}\right) \\ &\quad = p_{13*}''' (\operatorname {id}_{S_1}\times \operatorname {id}_{S_2}\times i_3)_*\\ &\qquad \times \left( (\operatorname {id}_{S_1}\times \operatorname {id}_{S_2}\times i_3)^* p_{12}^{\prime \prime \prime *} (i_1\times \operatorname {id}_{X_2})^* K_{12} \otimes ^L_{S_1\times S_2\times S_3} p_{23}^{\prime *} (i_2\times \operatorname {id}_{X_3})^* K_{23}\right) \\ &\quad = (\operatorname {id}_{S_1}\times i_3)_* p_{13*}' \left( p_{12}^{\prime *} (i_1\times \operatorname {id}_{X_2})^* K_{12} \otimes ^L_{S_1\times S_2\times S_3} p_{23}^{\prime *} (i_2\times \operatorname {id}_{X_3})^* K_{23}\right), \end{align*}$$

where we have used 6.5.1 in the second equality and $p_{13}'''\circ (\operatorname {id}_{S_1}\times \operatorname {id}_{S_2}\times i_3) = (\operatorname {id}_{S_1}\times i_3)\circ p_{13}'$, $p_{12}'''\circ (\operatorname {id}_{S_1}\times \operatorname {id}_{S_2}\times i_3) = p_{12}'$ in the third equality. Finally, by 8.2.2, the homomorphism $(\operatorname {id}_{S_1}\times i_3)_*$ is just the identity operator $K(Z_{12}'\circ Z_{23}') \to K(Z_{12}'\circ Z_{23}')$. Thus we have the assertion.

■

8.3. Convolution and principal bundles

Let $G$ be a linear algebraic group and suppose that we have principal $G$-bundles $\pi _a\colon P_a\to X_a$ over $X_a$ for $a=1, 2, 3$. Consider the restriction of the principal $G$-bundle $\pi _a\times \operatorname {id}_{X_b}\colon P_a\times X_b\to X_a\times X_b$ to $Z_{ab}$ for $(a,b) = (1,2), (2,3)$. Then the pull-back homomorphism gives a canonical isomorphism

$$\begin{equation} (\pi _a\times \operatorname {id}_{X_b})^* \colon K(Z_{ab}) \xrightarrow {\cong } K^G((\pi _a\times \operatorname {id}_{X_b})^{-1}Z_{ab}). {\tag{8.3.1}} \end{equation}$$

Similarly we have an isomorphism

$$\begin{equation} (\pi _1\times \operatorname {id}_{X_3})^* \colon K(Z_{12}\circ Z_{23}) \xrightarrow {\cong } K^G((\pi _1\times \operatorname {id}_{X_3})^{-1}(Z_{12}\circ Z_{23})). {\tag{8.3.2}} \end{equation}$$

Let $G$ act on $P_a\times P_b$ diagonally. We assume that there exists a closed $G$-invariant subvariety $Z_{ab}'$ of $P_a\times P_b$ such that

$$\begin{equation} \begin{split} & \text{the restriction of $\operatorname {id}_{P_a}\times \,\pi _b\colon P_a\times P_b\to P_a\times X_b$ to $Z_{ab}'$ is proper, and} \\ & (\operatorname {id}_{P_a}\times \,\pi _b)(Z_{ab}') = (\pi _a\times \operatorname {id}_{X_b})^{-1}Z_{ab} \end{split} {\tag{8.3.3}} \end{equation}$$

for $(a,b) = (1,2), (2,3)$.

Let $p_{ab}'$ be the projection $P_1\times P_2\times P_3\to P_a\times P_b$. Since the restriction of the projection $p_{13}'\colon p_{12}^{\prime -1}(Z_{12}')\cap p_{23}^{\prime -1}(Z_{23}')\to P_1\times P_3$ is proper, we have the convolution product $*'\colon K^G(Z_{12}')\otimes K^G(Z_{23}') \to K^G(Z_{12}'\circ Z_{23}')$ given by

$$\begin{equation*} K'_{12}*' K'_{23} \overset{\operatorname {\scriptstyle def.}}{=}p'_{13*} \left( p_{12}^{\prime *} K'_{12}\otimes ^L_{P_1\times P_2\times P_3} p_{23}^{\prime *} K'_{23}\right) \end{equation*}$$

for $K'_{12}\in K^G(Z_{12}')$, $K'_{23}\in K^G(Z_{23}')$. We want to compare this convolution product with that on $K(Z_{12})\otimes K(Z_{23})$.

By 8.3.3, we have

$$\begin{equation*} (\operatorname {id}_{P_a}\times \,\pi _b)_* K_{ab}' \in K^G((\operatorname {id}_{P_a}\times \,\pi _b)(Z_{ab}')) = K^G((\pi _a\times \operatorname {id}_{X_b})^{-1}Z_{ab}). \end{equation*}$$

Via 8.3.1, we define

$$\begin{equation} K_{ab} \overset{\operatorname {\scriptstyle def.}}{=}\left((\pi _a\times \operatorname {id}_{X_b})^*\right)^{-1} (\operatorname {id}_{P_a}\times \,\pi _b)_* K_{ab}' \in K(Z_{ab}). {\tag{8.3.4}} \end{equation}$$

Thus we can consider the convolution product $K_{12} * K_{23}$.

By the construction, we have

$$\begin{equation*} (\operatorname {id}_{P_1}\times \,\pi _3)(Z_{12}'\circ Z_{23}') = (\pi _1\times \operatorname {id}_{X_3})^{-1}(Z_{12}\circ Z_{23}). \end{equation*}$$

Noticing that the restriction of $\operatorname {id}_{P_1}\times \,\pi _3$ to $Z_{12}'\circ Z_{23}'$ is proper, we have

$$\begin{equation*} (\operatorname {id}_{P_1}\times \,\pi _3)_* (K_{12}' *' K_{23}') \in K^G((\pi _1\times \operatorname {id}_{X_3})^{-1}(Z_{12}\circ Z_{23}). \end{equation*}$$

Combining this with 8.3.2, we have

$$\begin{equation*} \left((\pi _1\times \operatorname {id}_{X_3})^*\right)^{-1}(\operatorname {id}_{P_1}\times \,\pi _3)_* (K_{12}' *' K_{23}') \in K(Z_{12}\circ Z_{23}). \end{equation*}$$

Proposition 8.3.5.

In the above setup, we have

$$\begin{equation} (\pi _1\times \operatorname {id}_{X_3})^* (K_{12} * K_{23}) = (\operatorname {id}_{P_1}\times \,\pi _3)_* ( K_{12}' *' K_{23}' ). {\tag{8.3.6}} \end{equation}$$

Namely the following diagram commutes:

$$\begin{equation*} \begin{CD} K^G(Z'_{12})\otimes K^G(Z'_{23}) @>{*'}>> K^G(Z'_{12}\circ Z'_{23}) \\@VVV @VVV \\K(Z_{12})\otimes K(Z_{23}) @>{*}>> K(Z_{12}\circ Z_{23}), \end{CD} \end{equation*}$$

where the left vertical arrow is

$$\begin{equation*} \left((\pi _1\times \operatorname {id}_{X_2})^*\right)^{-1} (\operatorname {id}_{P_1}\times \,\pi _2)_* \otimes \left((\pi _2\times \operatorname {id}_{X_3})^*\right)^{-1} (\operatorname {id}_{P_2}\times \,\pi _3)_* , \end{equation*}$$

and the right vertical arrow is

$$\begin{equation*} \left((\pi _1\times \operatorname {id}_{X_3})^*\right)^{-1} (\operatorname {id}_{P_1}\times \,\pi _3)_* . \end{equation*}$$

Example 8.3.7.

Suppose $X_1 = X_2$, $P_1 = P_2$ and $Z_{12} = \operatorname {Image}\Delta _{X_1}$, where $\Delta _{X_1}$ is the diagonal embedding $X_1\to X_1\times X_2$. If we take $Z_{12}' = \operatorname {Image}\Delta _{P_1}$, where $\Delta _{P_1}$ is the diagonal embedding $P_1\to P_1\times P_2$, the assumption 8.3.3 is satisfied. In fact, the restriction of $\operatorname {id}_{P_1}\times \,\pi _2$ to $Z_{12}'$ is an isomorphism. Take a vector bundle $E$ and consider $K_{12} = \Delta _{X_1*}[E]$. By the isomorphism $K(X_1) \xrightarrow [\cong ]{\pi _1^*} K^G(P_1)$, we can define $K_{12}' = \Delta _{P_1*}\pi _1^* [E]$. Then both $(\operatorname {id}_{P_1}\times \,\pi _2)_* K_{12}'$ and $(\pi _1\times \operatorname {id}_{X_2})^* K_{12}$ is $\Delta '_* [E]$ where $\Delta '\colon P_1\to (\operatorname {id}_{P_1}\times \,\pi _2)\Delta _{P_1} = (\pi _1\times \operatorname {id}_{X_2})^{-1}\Delta _{X_1}$ is the natural isomorphism. Hence 8.3.4 holds for $K_{12}$ and $K_{12}'$. By Lemma 8.1.1, we have

$$\begin{equation*} \begin{split} & (\Delta _{X_1})_*[E] * K_{23} = p_2^* [E] \otimes K_{23},\\ & (\Delta _{P_1})_*[\pi _1^* E] *' K_{23}' = p_2^{\prime *}\pi _2^* [E] \otimes K'_{23}, \end{split} \end{equation*}$$

where $p_2\colon X_2\times X_3\to X_2$, $p_2'\colon P_2\times X_3\to P_2$ are the projections. We can directly check 8.3.6 in this case.

Proof of Proposition 8.3.5.

As in the proof of Proposition 8.2.3, we replace $X_a$ by $P_a$ factor by factor.

Step 1. First we replace $X_1$ by $P_1$. Consider the following fiber square:

$$\begin{equation*} \begin{CD} P_1\times X_2\times X_3 @>{\pi _1\times \operatorname {id}_{X_2}\times \operatorname {id}_{X_3}}>> X_1\times X_2\times X_3 \\@V{p_{13}''}VV @VV{p_{13}}V \\P_1\times X_3 @>>{\pi _1\times \operatorname {id}_{X_3}}> X_1\times X_3, \end{CD} \end{equation*}$$

where $p_{13}''$ is the projection. By base change 13, 5.3.15 and 6.4.1, we have

$$\begin{equation} \begin{split} & (\pi _1\times \operatorname {id}_{X_3})^* (K_{12} * K_{23}) = (\pi _1\times \operatorname {id}_{X_3})^* p_{13*} \left( p_{12}^* K_{12}\otimes ^L_{X_1\times X_2\times X_3} p_{23}^* K_{23}\right) \\ &\quad = p_{13*}'' (\pi _1\times \operatorname {id}_{X_2}\times \operatorname {id}_{X_3})^* \left( p_{12}^* K_{12}\otimes ^L_{X_1\times X_2\times X_3} p_{23}^* K_{23}\right) \\ &\quad = p_{13*}'' \left( (\pi _1\times \operatorname {id}_{X_2}\times \operatorname {id}_{X_3})^* p_{12}^* K_{12} \otimes ^L_{P_1\times X_2\times X_3} (\pi _1\times \operatorname {id}_{X_2}\times \operatorname {id}_{X_3})^* p_{23}^* K_{23} \right) \\ &\quad = p_{13*}'' \left( p_{12}^{\prime \prime *}(\pi _1\times \operatorname {id}_{X_2})^* K_{12} \otimes ^L_{P_1\times X_2\times X_3} p_{23}^{\prime \prime *} K_{23} \right), \end{split} {\tag{8.3.8}} \end{equation}$$

where $p_{12}''\colon P_1\times X_2\times X_3\to P_1\times X_2$ and $p_{23}''\colon P_1\times X_2\times X_3\to X_2\times X_3$ are projections.

Step 2. Consider the fiber square

$$\begin{equation*} \begin{CD} P_1\times P_2\times X_3 @>{p_{12}'''}>> P_1\times P_2 \\@V{\operatorname {id}_{P_1}\times \,\pi _2\times \operatorname {id}_{X_3}}VV @VV{\operatorname {id}_{P_1}\times \,\pi _2}V \\P_1\times X_2\times X_3 @>>{p_{12}''}> P_1\times X_2, \end{CD} \end{equation*}$$

where $p_{12}'''$ is the projection. By base change 13, 5.3.15, we have

$$\begin{equation*} (\operatorname {id}_{P_1}\times \,\pi _2\times \operatorname {id}_{X_3})_* p_{12}^{\prime \prime \prime *} K_{12}' = p_{12}^{\prime \prime *} (\operatorname {id}_{P_1}\times \pi _2)_* K_{12}' = p_{12}^{\prime \prime *}(\pi _1\times \operatorname {id}_{X_2})^* K_{12}, \end{equation*}$$

where we have used 8.3.4 for $(a,b) = (1,2)$. Substituting this into 8.3.8, we get

$$\begin{equation} \begin{split} & (\pi _1\times \operatorname {id}_{X_3})^* (K_{12} * K_{23}) \\ &\qquad = p_{13*}''\left( (\operatorname {id}_{P_1}\times \,\pi _2\times \operatorname {id}_{X_3})_* p_{12}^{\prime \prime \prime *} K_{12}' \otimes ^L_{P_1\times X_2\times X_3} p_{23}^{\prime \prime *} K_{23}\right) \\ &\qquad = p_{13*}'' (\operatorname {id}_{P_1}\times \,\pi _2\times \operatorname {id}_{X_3})_* \left( p_{12}^{\prime \prime \prime *} K_{12}' \otimes ^L_{P_1\times P_2\times X_3} (\operatorname {id}_{P_1}\times \,\pi _2\times \operatorname {id}_{X_3})^* p_{23}^{\prime \prime *}K_{23}\right), \end{split} {\tag{8.3.9}} \end{equation}$$

where we have used 6.5.1 in the second equality. Let $p_{23}'''\colon P_1\times P_2\times X_3\to P_2\times X_3$ be the projection. By $p_{23}''\circ (\operatorname {id}_{P_1}\times \,\pi _2\times \operatorname {id}_{X_3}) = (\pi _2\times \operatorname {id}_{X_3})\circ p_{23}'''$, we have

$$\begin{equation*} (\operatorname {id}_{P_1}\times \,\pi _2\times \operatorname {id}_{X_3})^* p_{23}^{\prime \prime *} = p_{23}^{\prime \prime \prime *} (\pi _2\times \operatorname {id}_{X_3})^*. \end{equation*}$$

We also have

$$\begin{equation*} p_{13*}'' (\operatorname {id}_{P_1}\times \,\pi _2\times \operatorname {id}_{X_3})_* = p_{13*}''', \end{equation*}$$

where $p_{13}\colon P_1\times P_2\times X_3 \to P_1\times X_3$ is the projection. Substituting these two equalities into 8.3.9, we get

$$\begin{equation} (\pi _1\times \operatorname {id}_{X_3})^* (K_{12} * K_{23}) = p_{13*}'''\left( p_{12}^{\prime \prime \prime *} K_{12}' \otimes ^L_{P_1\times P_2\times X_3} p_{23}^{\prime \prime \prime *} (\pi _2\times \operatorname {id}_{X_3})^* K_{23}\right). {\tag{8.3.10}} \end{equation}$$

Step 3. Consider the fiber square

$$\begin{equation*} \begin{CD} P_1\times P_2\times P_3 @>{p_{23}'}>> P_2\times P_3\\@V{\operatorname {id}_{P_1}\times \operatorname {id}_{P_2}\times \,\pi _3}VV @VV{\operatorname {id}_{P_2}\times \,\pi _3}V \\P_1\times P_2\times X_3 @>{p_{23}'''}>> P_2\times X_3. \end{CD} \end{equation*}$$

By base change 13, 5.3.15, we have

$$\begin{equation*} (\operatorname {id}_{P_1}\times \operatorname {id}_{P_2}\times \,\pi _3)_* p_{23}^{\prime *} K'_{23} = p_{23}^{\prime \prime \prime *} (\operatorname {id}_{P_2}\times \,\pi _3)_* K'_{23} = p_{23}^{\prime \prime \prime *} (\pi _2\times \operatorname {id}_{X_3})^* K_{23}, \end{equation*}$$

where we have used 8.3.4 for $(a,b) = (2,3)$ in the second equality. Substituting this into 8.3.10, we have

$$\begin{align*} & (\pi _1\times \operatorname {id}_{X_3})^* (K_{12} * K_{23}) \\ &\qquad = p_{13*}'''\left( p_{12}^{\prime \prime \prime *} K_{12}' \otimes ^L_{P_1\times P_2\times X_3} (\operatorname {id}_{P_1}\times \operatorname {id}_{P_2}\times \,\pi _3)_* p_{23}^{\prime *}K_{23}' \right) \\ &\qquad = p_{13*}''' (\operatorname {id}_{P_1}\times \operatorname {id}_{P_2}\times \,\pi _3)_* \left( (\operatorname {id}_{P_1}\times \operatorname {id}_{P_2}\times \,\pi _3)^* p_{12}^{\prime \prime \prime *} K_{12}' \otimes ^L_{P_1\times P_2\times P_3} p_{23}^{\prime *} K_{23}'\right). \end{align*}$$

By $p_{13}'''\circ (\operatorname {id}_{P_1}\times \operatorname {id}_{P_2}\times \,\pi _3) = (\operatorname {id}_{P_1}\times \,\pi _3)\circ p_{13}'$ and $p_{12}'''\circ (\operatorname {id}_{P_1}\times \operatorname {id}_{P_2}\times \,\pi _3) = p_{12}'$, we get

$$\begin{equation*} (\pi _1\times \operatorname {id}_{X_3})^* (K_{12} * K_{23}) = (\operatorname {id}_{P_1}\times \,\pi _3)_*\circ p_{13*}'\left( p_{12}^{\prime *} K_{12}' \otimes ^L_{P_1\times P_2\times P_3} p_{23}^{\prime *} K_{23}'\right). \end{equation*}$$

This proves our assertion.

■

9. A homomorphism ${\mathbf{U}}_q({\mathbf{L}}{\mathfrak{g}})\to K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(Z({\mathbf{w}}))\otimes _{{\mathbb{Z}}[q,q^{-1}]}{\mathbb{Q}}(q)$

9.1

Let us define an analogue of the Steinberg variety $Z({\mathbf{v}}^1,{\mathbf{v}}^2;{\mathbf{w}})$ by

$$\begin{equation} \{ (x^1, x^2)\in {\mathfrak{M}}({\mathbf{v}}^1,{\mathbf{w}})\times {\mathfrak{M}}({\mathbf{v}}^2,{\mathbf{w}}) \mid \pi (x^1) = \pi (x^2) \}. {\tag{9.1.1}} \end{equation}$$

Here $\pi (x^1) = \pi (x^2)$ means that $\pi (x^1)$ is equal to $\pi (x^2)$ if we regard both as elements of ${\mathfrak{M}}_0(\infty ,{\mathbf{w}})$ by 2.5.4. This is a closed subvariety of ${\mathfrak{M}}({\mathbf{v}}^1,{\mathbf{w}})\times {\mathfrak{M}}({\mathbf{v}}^2,{\mathbf{w}})$.

The map $p_{13}\colon p_{12}^{-1}(Z({\mathbf{v}}^1,{\mathbf{v}}^2;{\mathbf{w}}))\cap p_{23}^{-1}(Z({\mathbf{v}}^2,{\mathbf{v}}^3;{\mathbf{w}})) \to {\mathfrak{M}}({\mathbf{v}}^1,{\mathbf{w}})\times {\mathfrak{M}}({\mathbf{v}}^3,{\mathbf{w}})$ is proper and its image is contained in $Z({\mathbf{v}}^1,{\mathbf{v}}^3;{\mathbf{w}})$. Hence we can define the convolution product on the equivariant $K$-theory:

$$\begin{equation*} K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(Z({\mathbf{v}}^1,{\mathbf{v}}^2;{\mathbf{w}}))\otimes K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(Z({\mathbf{v}}^2,{\mathbf{v}}^3;{\mathbf{w}})) \to K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(Z({\mathbf{v}}^1,{\mathbf{v}}^3;{\mathbf{w}})). \end{equation*}$$

Let

$$\begin{equation*} {\prod _{{\mathbf{v}}^1,{\mathbf{v}}^2}}' K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(Z({\mathbf{v}}^1,{\mathbf{v}}^2;{\mathbf{w}})) \end{equation*}$$

be the subspace of $\prod _{{\mathbf{v}}^1,{\mathbf{v}}^2} K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(Z({\mathbf{v}}^1,{\mathbf{v}}^2;{\mathbf{w}}))$ consisting of elements $(F_{{\mathbf{v}}^1,{\mathbf{v}}^2})$ such that

(1)

for fixed ${\mathbf{v}}^1$, $F_{{\mathbf{v}}^1,{\mathbf{v}}^2} = 0$ for all but finitely many choices of ${\mathbf{v}}^2$,

(2)

for fixed ${\mathbf{v}}^2$, $F_{{\mathbf{v}}^1,{\mathbf{v}}^2} = 0$ for all but finitely many choices of ${\mathbf{v}}^1$.

The convolution product $*$ is well defined on $\prod '_{{\mathbf{v}}^1,{\mathbf{v}}^2} K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(Z({\mathbf{v}}^1,{\mathbf{v}}^2;{\mathbf{w}}))$. When the underlying graph is of type $ADE$, ${\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})$ is empty for all but finitely many choices of ${\mathbf{v}}$, so $\prod '$ is just the direct product $\prod$.

Let $Z({\mathbf{w}})$ denote the disjoint union $\bigsqcup _{{\mathbf{v}}^1,{\mathbf{v}}^2} Z({\mathbf{v}}^1,{\mathbf{v}}^2;{\mathbf{w}})$. When we write

$$\begin{equation*} K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(Z({\mathbf{w}})), \end{equation*}$$

we mean $\prod '_{{\mathbf{v}}^1,{\mathbf{v}}^2} K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(Z({\mathbf{v}}^1,{\mathbf{v}}^2;{\mathbf{w}}))$ as convention.

The second projection $G_{\mathbf{w}}\times {\mathbb{C}}^*\to {\mathbb{C}}^*$ induces a homomorphism $R({\mathbb{C}}^*)\to R(G_{\mathbf{w}}\times {\mathbb{C}}^*)$. Thus $R(G_{\mathbf{w}}\times {\mathbb{C}}^*)$ is an $R({\mathbb{C}}^*)$-algebra. Moreover, $R({\mathbb{C}}^*)$ is isomorphic to ${\mathbb{Z}}[q,q^{-1}]$ where $q^m$ corresponds to $L(m)$ in 2.8.1. Thus $K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(Z({\mathbf{w}}))$ is a ${\mathbb{Z}}[q,q^{-1}]$-algebra.

The aim of this section and the next two sections is to define the homomorphism from ${\mathbf{U}}_q({\mathbf{L}}{\mathfrak{g}})$ into $K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(Z({\mathbf{w}}))\otimes _{{\mathbb{Z}}[q,q^{-1}]}{\mathbb{Q}}(q)$. We first define the map on generators of ${\mathbf{U}}_q({\mathbf{L}}{\mathfrak{g}})$, and then check the defining relation.

9.2

First we want to define the image of $q^h$, $h_{k,m}$.

Let $C_k^\bullet ({\mathbf{v}},{\mathbf{w}})$ be the $G_{\mathbf{w}}\times {\mathbb{C}}^*$-equivariant complex over ${\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})$ defined in 2.9.1. We consider $C_k^\bullet ({\mathbf{v}},{\mathbf{w}})$ as an element of $K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}({\mathfrak{M}}({\mathbf{v}},{\mathbf{w}}))$ by identifying it with the alternating sum

$$\begin{equation*} - [q^{-2} V_k] + \left[q^{-1} \left( \displaystyle {\bigoplus _{l:k\neq l}} [-\langle h_k,\alpha _l\rangle ]_q V_l \oplus W_k\right) \right] - [V_k]. \end{equation*}$$

The rank of the complex 2.9.1, as an element of $K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}({\mathfrak{M}}({\mathbf{v}},{\mathbf{w}}))$ (see §6.2), is given by

$$\begin{equation*} \operatorname {rank}C^\bullet _k({\mathbf{v}},{\mathbf{w}}) = - \sum _{l:k\neq l} (\alpha _k,\alpha _l) \dim V_l + \dim W_k - 2\dim V_k = \langle h_k, {\mathbf{w}}- {\mathbf{v}}\rangle . \end{equation*}$$

Let $\Delta$ denote the diagonal embedding ${\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})\to {\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})\times {\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})$;

$$\begin{equation} \begin{gathered} q^{h} \longmapsto \sum _{\mathbf{v}}q^{\langle h, {\mathbf{w}}-{\mathbf{v}}\rangle } \Delta _* \mathcal{O}_{{\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})}, \\ p_k^+(z) \longmapsto \sum _{{\mathbf{v}}} \Delta _* \left({\textstyle \bigwedge }_{-1/z} (C_k^\bullet ({\mathbf{v}},{\mathbf{w}}))\right)^+, \\ p_k^-(z) \longmapsto \sum _{{\mathbf{v}}} \Delta _* \left((-z)^{\operatorname {rank}C_k^\bullet ({\mathbf{v}},{\mathbf{w}})}\det C_k^\bullet ({\mathbf{v}},{\mathbf{w}})^* {\textstyle \bigwedge }_{-1/z} (C_k^\bullet ({\mathbf{v}},{\mathbf{w}}))\right)^-, \end{gathered} {\tag{9.2.1}} \end{equation}$$ where $(\ )^\pm$ denotes the expansion at $z = \infty$, $0$ respectively. Note that

$$\begin{equation*} \psi _k^\pm (z) = q^{\pm h_k} \frac{p_k^\pm (q z)}{p_k^\pm (q^{-1}z)} \longmapsto \sum _{{\mathbf{v}}} q^{\operatorname {rank}C_k^\bullet ({\mathbf{v}},{\mathbf{w}})}\Delta _* \left(\frac{{\textstyle \bigwedge }_{-1/qz} (C_k^\bullet ({\mathbf{v}},{\mathbf{w}}))}{{\textstyle \bigwedge }_{-q/z} (C_k^\bullet ({\mathbf{v}},{\mathbf{w}}))}\right)^\pm . \end{equation*}$$

9.3

Next we define the images of $e_{k,r}$ and $f_{k,r}$. They are given by line bundles over Hecke correspondences.

Let ${\mathbf{v}}^1$, ${\mathbf{v}}^2$ and ${\mathfrak{P}}_k({\mathbf{v}}^2,{\mathbf{w}})$ be as in §5.1. By the definition, the quotient $V_k^2/V_k^1$ defines a line bundle over ${\mathfrak{P}}_k({\mathbf{v}}^2,{\mathbf{w}})$. The generator $e_{k,r}$ is very roughly defined as the $r$th power of $V_k^2/V_k^1$, but we need a certain modification in order to have the correct commutation relation.

For the modification, we need to consider the following variants of $C^\bullet _k({\mathbf{v}},{\mathbf{w}})$:

$$\begin{equation} C^{\prime \bullet }_k({\mathbf{v}},{\mathbf{w}}) \overset{\operatorname {\scriptstyle def.}}{=}\operatorname {Coker}\sigma _k , \qquad C^{\prime \prime \bullet }_k({\mathbf{v}},{\mathbf{w}}) \overset{\operatorname {\scriptstyle def.}}{=}V_k[-1], {\tag{9.3.1}} \end{equation}$$

where $V_k[-1]$ means that we consider the complex consisting of $V_k$ in degrees $1$ and $0$ for other degrees. Since $\alpha$ is injective, we have $C^\bullet _k({\mathbf{v}},{\mathbf{w}}) = C^{\prime \bullet }_k({\mathbf{v}},{\mathbf{w}}) + C^{\prime \prime \bullet }_k({\mathbf{v}},{\mathbf{w}})$ in $K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}({\mathfrak{M}}({\mathbf{v}},{\mathbf{w}}))$. We have the corresponding decomposition of the Cartan matrix ${\mathbf{C}}$:

$$\begin{equation*} {\mathbf{C}}= {\mathbf{C}}' + {\mathbf{C}}'', \qquad \text{where ${\mathbf{C}}' \overset{\operatorname {\scriptstyle def.}}{=}{\mathbf{C}}- {\mathbf{I}}$, ${\mathbf{C}}'' \overset{\operatorname {\scriptstyle def.}}{=}{\mathbf{I}}$.} \end{equation*}$$

We identify ${\mathbf{C}}'$ (resp. ${\mathbf{C}}''$) with a map given by

$$\begin{equation*} {\mathbf{v}}= \sum v_k \alpha _k \in \bigoplus {\mathbb{Z}}\alpha _k \mapsto \sum v_k (\alpha _k - \Lambda _k) \quad \left(\text{resp. }\sum v_k \Lambda _k\right) \in P. \end{equation*}$$

We also need matrices ${\mathbf{C}}_\Omega$, ${\mathbf{C}}_{\overline{\Omega }}$ given by

$$\begin{equation*} {\mathbf{C}}_\Omega \overset{\operatorname {\scriptstyle def.}}{=}{\mathbf{I}}- {\mathbf{A}}_\Omega , \qquad {\mathbf{C}}_{\overline{\Omega }} \overset{\operatorname {\scriptstyle def.}}{=}{\mathbf{I}}- {\mathbf{A}}_{\overline{\Omega }}, \end{equation*}$$

where ${\mathbf{A}}_\Omega$, ${\mathbf{A}}_{\overline{\Omega }}$ are as in 2.1.1. We also identify them with maps $\bigoplus {\mathbb{Z}}\alpha _k \to P$ exactly as above.

Let $\omega \colon {\mathfrak{M}}({\mathbf{v}}^1,{\mathbf{w}})\times {\mathfrak{M}}({\mathbf{v}}^2,{\mathbf{w}}) \to {\mathfrak{M}}({\mathbf{v}}^2,{\mathbf{w}})\times {\mathfrak{M}}({\mathbf{v}}^1,{\mathbf{w}})$ be the exchange of factors. Let us denote $\omega \left({\mathfrak{P}}_k({\mathbf{v}}^2+\alpha _k,{\mathbf{w}})\right) \subset {\mathfrak{M}}({\mathbf{v}}^2+\alpha _k,{\mathbf{w}})\times {\mathfrak{M}}({\mathbf{v}}^2,{\mathbf{w}})$ by ${\mathfrak{P}}_k^-({\mathbf{v}}^2,{\mathbf{w}})$. As on ${\mathfrak{P}}_k({\mathbf{v}},{\mathbf{w}})$, we have a natural line bundle over ${\mathfrak{P}}_k^-({\mathbf{v}}^2,{\mathbf{w}})$. Let us denote it by $V^1_k/V^2_k$.

Now we define the images of $e_{k,r}$, $f_{k,r}$ by

$$\begin{equation} \begin{split} & e_{k,r} \longmapsto \sum _{{\mathbf{v}}^2} (-1)^{-\langle h_k, {\mathbf{C}}_{\overline{\Omega }}{\mathbf{v}}^2\rangle } \left[ i_* \left(q^{-1}V_k^2/V_k^1\right)^{\otimes r-\langle h_k, {\mathbf{C}}''{\mathbf{v}}^2\rangle } \otimes \det C_k^{\prime \bullet }({\mathbf{v}}^2,{\mathbf{w}})^*\right], \\ & f_{k,r} \longmapsto \sum _{{\mathbf{v}}^2} (-1)^{\langle h_k, {\mathbf{w}}- {\mathbf{C}}_\Omega {\mathbf{v}}^2\rangle } \left[ i^-_* \left(q^{-1}V_k^1/V_k^2\right)^{\otimes r+\langle h_k, {\mathbf{w}}-{\mathbf{C}}'{\mathbf{v}}^2\rangle } \otimes \det C_k^{\prime \prime \bullet }({\mathbf{v}}^2,{\mathbf{w}})^*\right], \end{split} {\tag{9.3.2}} \end{equation}$$

where $i\colon {\mathfrak{P}}_k({\mathbf{v}}^2,{\mathbf{w}})\to Z({\mathbf{v}}^2-\alpha _k,{\mathbf{v}}^2;{\mathbf{w}})$ and $i^-\colon {\mathfrak{P}}^-_k({\mathbf{v}}^2,{\mathbf{w}})\to Z({\mathbf{v}}^2+\alpha _k,{\mathbf{v}}^2;{\mathbf{w}})$ are the inclusions. Hereafter, we may omit $i_*$ or $i^-_*$, hoping that it causes no confusion.

9.4

Theorem 9.4.1.

The assignments 9.2.1, 9.3.2 define a homomorphism

$$\begin{equation*} {\mathbf{U}}_q({\mathbf{L}}{\mathfrak{g}})\to K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(Z({\mathbf{w}}))\otimes _{{\mathbb{Z}}[q,q^{-1}]}{\mathbb{Q}}(q) \end{equation*}$$

of ${\mathbb{Q}}(q)$-algebras.

We need to check the defining relations 1.2.1, 1.2.2, 1.2.3, 1.2.4, 1.2.5, 1.2.6, 1.2.7, 1.2.8, 1.2.9, 1.2.10, and 1.2.11. We do not need to consider the relations 1.2.1, 1.2.5 because we are considering ${\mathbf{U}}_q({\mathbf{L}}{\mathfrak{g}})$ instead of ${\mathbf{U}}_q(\widehat{\mathfrak{g}})$. The relations 1.2.2, 1.2.3 and 1.2.4 follow from Lemma 8.1.1 and the fact that $E \otimes F\cong F\otimes E$. The relation 1.2.6 also follows from Lemma 8.1.1. The remaining relations will be checked in the next two sections.

10. Relations (I)

10.1. Relation 1.2.7

Fix a vertex $k\in I$ and take ${\mathbf{v}}^1$, ${\mathbf{v}}^2 = {\mathbf{v}}^1+\alpha _k$. Let $i$ be the inclusion ${\mathfrak{P}}_k({\mathbf{v}}^2,{\mathbf{w}})\to Z({\mathbf{v}}^1,{\mathbf{v}}^2;{\mathbf{w}})$ and let $p_1$ and $p_2$ be the projections ${\mathfrak{P}}_k({\mathbf{v}}^2,{\mathbf{w}})\to {\mathfrak{M}}({\mathbf{v}}^1,{\mathbf{w}})$ and ${\mathfrak{P}}_k({\mathbf{v}}^2,{\mathbf{w}})\to {\mathfrak{M}}({\mathbf{v}}^2,{\mathbf{w}})$ respectively.

By Lemma 8.1.1, we have

$$\begin{equation} \begin{split} & \Delta _* {\textstyle \bigwedge }_{-1/z}(C_l^\bullet ({\mathbf{v}}^1,{\mathbf{w}}))\ast i_{*} \left[\sum _{r=-\infty }^\infty \left(q^{-1} V^2_k/V^1_k\right)^{\otimes r} w^{-r} \right] \ast \Delta _* \left({\textstyle \bigwedge }_{-1/z} C_l^\bullet ({\mathbf{v}}^2,{\mathbf{w}})\right)^{-1}\\ &= i_* \left[{\textstyle \bigwedge }_{-1/z} p_1^* (C_l^\bullet ({\mathbf{v}}^1,{\mathbf{w}}))\otimes \left({\textstyle \bigwedge }_{-1/z} p_2^* (C_l^\bullet ({\mathbf{v}}^2,{\mathbf{w}}))\right)^{-1}\otimes \sum _{r=-\infty }^\infty \left(q^{-1}V^2_k/V^1_k\right)^{\otimes r} w^{-r}\right]. \end{split} {\tag{10.1.1}} \end{equation}$$

We have the following equality in $K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}({\mathfrak{P}}_k({\mathbf{v}}^2,{\mathbf{w}}))$:

$$\begin{equation*} V_k^1 = V_k^2 - V_k^2/V_k^1. \end{equation*}$$

Hence we have

$$\begin{equation*} {\textstyle \bigwedge }_{-1/z} p_1^* C_l^\bullet ({\mathbf{v}}^1,{\mathbf{w}})\otimes \left({\textstyle \bigwedge }_{-1/z} p_2^* C_l^\bullet ({\mathbf{v}}^2,{\mathbf{w}})\right)^{-1} = {\textstyle \bigwedge }_{-1/z} [\langle h_k,\alpha _l\rangle ]_q \left(q^{-1}V^2_k/V^1_k\right). \end{equation*}$$

Substituting this into 10.1.1, we get

$$\begin{align*} & \Delta _* {\textstyle \bigwedge }_{-1/z} (C_l^\bullet ({\mathbf{v}}^1,{\mathbf{w}}))\ast i_{*} \left[\sum _{r=-\infty }^\infty \left(q^{-1} V^2_k/V^1_k\right)^{\otimes r} w^{-r} \right] \ast \Delta _* \left({\textstyle \bigwedge }_{-1/z} C_l^\bullet ({\mathbf{v}}^2,{\mathbf{w}})\right)^{-1} \\ & = \begin{cases} {\displaystyle \left(1 - \frac{wq}{z}\right)\left(1 - \frac{w}{zq}\right) i_* \left[\sum _{r=-\infty }^\infty \left(q^{-1}V^2_k/V^1_k\right)^{\otimes r} w^{-r}\right]} & \text{if $k = l$,} \\ {\displaystyle \prod _{p=0}^{-\langle h_k,\alpha _l\rangle -1} \left(1 - \frac{q^{\langle h_k,\alpha _l\rangle +1+2p}w}{z}\right)^{-1} i_* \left[\sum _{r=-\infty }^\infty \left(q^{-1}V^2_k/V^1_k\right)^{\otimes r} w^r\right]} & \text{otherwise.} \end{cases} \\ \end{align*}$$

This is equivalent to 1.2.7 for $x_l^+(w)$. The relation 1.2.7 for $x_l^-(w)$ can be proved in the same way.

10.2. Relation 1.2.8 for $k\neq l$

Fix two vertices $k\neq l$. Let ${\mathbf{v}}^1$, ${\mathbf{v}}^2$, ${\mathbf{v}}^3$, ${\mathbf{v}}^4$ be dimension vectors such that

$$\begin{equation*} {\mathbf{v}}^2 = {\mathbf{v}}^1 + \alpha _k = {\mathbf{v}}^3 + \alpha _l, \quad {\mathbf{v}}^4 = {\mathbf{v}}^1 - \alpha _l = {\mathbf{v}}^3 - \alpha _k. \end{equation*}$$

We want to compute $e_{k,r} * f_{l,s}$ and $f_{l,s} * e_{k,r}$ in the component $K^{G_{\mathbf{w}}\times {\mathbb{C}}}(Z({\mathbf{v}}^1,{\mathbf{v}}^3,{\mathbf{w}}))$.

Let us consider the intersection

$$\begin{equation*} p_{12}^{-1}{\mathfrak{P}}_k({\mathbf{v}}^2,{\mathbf{w}}) \cap p_{23}^{-1}{\mathfrak{P}}^-_{l}({\mathbf{v}}^3,{\mathbf{w}}) \qquad \text{(resp. $p_{12}^{-1}{\mathfrak{P}}^-_{l}({\mathbf{v}}^2,{\mathbf{w}}) \cap p_{23}^{-1}{\mathfrak{P}}_{k}({\mathbf{v}}^3,{\mathbf{w}})$)} \end{equation*}$$

in ${\mathfrak{M}}({\mathbf{v}}^1,{\mathbf{w}})\times {\mathfrak{M}}({\mathbf{v}}^2,{\mathbf{w}})\times {\mathfrak{M}}({\mathbf{v}}^3,{\mathbf{w}})$ (resp. ${\mathfrak{M}}({\mathbf{v}}^1,{\mathbf{w}})\times {\mathfrak{M}}({\mathbf{v}}^4,{\mathbf{w}})\times {\mathfrak{M}}({\mathbf{v}}^3,{\mathbf{w}})$). On the intersection, we have the inclusion of restrictions of tautological bundles

$$\begin{equation*} V^1 \subset V^2 \supset V^3 \qquad \text{(resp. $V^1 \supset V^4 \subset V^3$)}. \end{equation*}$$

Lemma 10.2.1.

The above two intersections are transversal, and there is a $G_{\mathbf{w}}\times {\mathbb{C}}^*$-equivariant isomorphism between them such that

(a)

it is the identity operators on the factor ${\mathfrak{M}}({\mathbf{v}}^1,{\mathbf{w}})$ and ${\mathfrak{M}}({\mathbf{v}}^3,{\mathbf{w}})$,

(b)

it induces isomorphisms $V^2_k/V^1_k\cong V^3_k/V^4_k$ and $V^2_l/V^3_l\cong V^1_l/V^4_l$.

Proof.

See 45, Lemmas 9.8, 9.9, 9.10 and their proofs.

■

Since the intersection is transversal, we have

$$\begin{equation} e_{k,r} * f_{l,s} = (-1)^{-\langle h_k, {\mathbf{C}}_{\overline{\Omega }}{\mathbf{v}}^2\rangle +\langle h_l, {\mathbf{w}}-{\mathbf{C}}_\Omega {\mathbf{v}}^3\rangle } p_{13*}[\mathcal{L}], {\tag{10.2.2}} \end{equation}$$

where $\mathcal{L}$ is the following line bundle over $p_{12}^{-1}{\mathfrak{P}}_k({\mathbf{v}}^2,{\mathbf{w}}) \cap p_{23}^{-1}{\mathfrak{P}}^-_{l}({\mathbf{v}}^3,{\mathbf{w}})$:

$$\begin{multline*} \left(q^{-1}V_k^2/V^1_k\right)^{\otimes r-\langle h_k, {\mathbf{C}}''{\mathbf{v}}^2\rangle } \otimes \left(q^{-1}V_l^2/V^3_l\right)^{\otimes s+\langle h_l, {\mathbf{w}}-{\mathbf{C}}'{\mathbf{v}}^3\rangle }\\ \otimes \det C_k^{\prime \bullet }({\mathbf{v}}^2,{\mathbf{w}})^* \otimes \det C_l^{\prime \prime \bullet }({\mathbf{v}}^3,{\mathbf{w}})^*. \end{multline*}$$

Similarly, we have

$$\begin{equation} f_{l,s} * e_{k,r} = (-1)^{\langle h_l, {\mathbf{w}}-{\mathbf{C}}_\Omega {\mathbf{v}}^4\rangle -\langle h_k, {\mathbf{C}}_{\overline{\Omega }}{\mathbf{v}}^3\rangle } p_{13*}[\mathcal{L}'], {\tag{10.2.3}} \end{equation}$$

where $\mathcal{L}'$ is the following line bundle over $p_{12}^{-1}{\mathfrak{P}}^-_{l}({\mathbf{v}}^2,{\mathbf{w}}) \cap p_{23}^{-1}{\mathfrak{P}}_{k}({\mathbf{v}}^3,{\mathbf{w}})$:

$$\begin{multline*} \left(q^{-1}V_l^1/V^4_l\right)^{\otimes s+\langle h_l, {\mathbf{w}}-{\mathbf{C}}'{\mathbf{v}}^4\rangle } \otimes \left(q^{-1}V_k^3/V^4_k\right)^{\otimes r-\langle h_k, {\mathbf{C}}''{\mathbf{v}}^3\rangle }\\ \otimes \det C_l^{\prime \prime \bullet }({\mathbf{v}}^4,{\mathbf{w}})^* \otimes \det C_k^{\prime \bullet }({\mathbf{v}}^3,{\mathbf{w}})^*. \end{multline*}$$

Let us compare 10.2.2 and 10.2.3. On $p_{12}^{-1}{\mathfrak{P}}_k({\mathbf{v}}^2,{\mathbf{w}}) \cap p_{23}^{-1}{\mathfrak{P}}^-_{l}({\mathbf{v}}^3,{\mathbf{w}})$, we have

$$\begin{equation*} \begin{split} \det C_k^{\prime \bullet }({\mathbf{v}}^2,{\mathbf{w}}) & = \det C_k^{\prime \bullet }({\mathbf{v}}^3,{\mathbf{w}}) \otimes \det \left([-(\alpha _k,\alpha _l)]_q (q^{-1}V^2_l/V^3_l)\right) \\ & = \det C_k^{\prime \bullet }({\mathbf{v}}^3,{\mathbf{w}}) \otimes \left(q^{-1}V^2_l/V^3_l\right)^{\otimes -(\alpha _k,\alpha _l)}. \end{split} \end{equation*}$$

On the other hand, we have

$$\begin{equation*} \det C_l^{\prime \prime \bullet }({\mathbf{v}}^3,{\mathbf{w}}) = \det C_l^{\prime \prime \bullet }({\mathbf{v}}^4,{\mathbf{w}}) \end{equation*}$$

on $p_{12}^{-1}{\mathfrak{P}}^-_l({\mathbf{v}}^2,{\mathbf{w}}) \cap p_{23}^{-1}{\mathfrak{P}}_{k}({\mathbf{v}}^3,{\mathbf{w}})$. Hence under the isomorphism in Lemma 10.2.1, we obtain

$$\begin{equation*} \mathcal{L} \cong \mathcal{L}', \end{equation*}$$

where we have used

$$\begin{equation} \langle h_l, {\mathbf{C}}'{\mathbf{v}}^3\rangle = \langle h_l, {\mathbf{C}}'{\mathbf{v}}^4\rangle - {\mathbf{A}}_{lk}, \quad \langle h_k, {\mathbf{C}}''{\mathbf{v}}^2\rangle = \langle h_k, {\mathbf{C}}''{\mathbf{v}}^3\rangle . {\tag{10.2.4}} \end{equation}$$

By

$$\begin{align*} &\langle h_l, {\mathbf{C}}_\Omega {\mathbf{v}}^3\rangle = \langle h_l, {\mathbf{C}}_\Omega {\mathbf{v}}^4\rangle - ({\mathbf{A}}_\Omega )_{lk}, \\ & \langle h_k, {\mathbf{C}}_{\overline{\Omega }}{\mathbf{v}}^2\rangle = \langle h_k, {\mathbf{C}}_{\overline{\Omega }}{\mathbf{v}}^3\rangle - ({\mathbf{A}}_{\overline{\Omega }})_{kl}, \quad ({\mathbf{A}}_\Omega )_{lk} = ({\mathbf{A}}_{\overline{\Omega }})_{kl}, \end{align*}$$

we have

$$\begin{equation*} (-1)^{-\langle h_k, {\mathbf{C}}_{\overline{\Omega }}{\mathbf{v}}^2\rangle +\langle h_l, {\mathbf{w}}-{\mathbf{C}}_\Omega {\mathbf{v}}^3\rangle } = (-1)^{\langle h_l, {\mathbf{w}}-{\mathbf{C}}_\Omega {\mathbf{v}}^4\rangle -\langle h_k, {\mathbf{C}}_{\overline{\Omega }}{\mathbf{v}}^3\rangle }. \end{equation*}$$

Thus we have $[e_{k,r}, f_{l,s}] = 0$.

10.3. Relation 1.2.10

We give the proof of 1.2.10 for $\pm = +$ in this subsection. The relation 1.2.10 for $\pm = -$ can be proved in a similar way, and hence is omitted.

Fix two vertices $k\neq l$. Let ${\mathbf{v}}^1$, ${\mathbf{v}}^2$, ${\mathbf{v}}^3$, ${\mathbf{v}}^4$ be dimension vectors such that

$$\begin{equation*} {\mathbf{v}}^2 = {\mathbf{v}}^1 + \alpha _k, \quad {\mathbf{v}}^4 = {\mathbf{v}}^1 + \alpha _l, \quad {\mathbf{v}}^3 = {\mathbf{v}}^2 + \alpha _l = {\mathbf{v}}^4 + \alpha _k = {\mathbf{v}}^1 + \alpha _k + \alpha _l. \end{equation*}$$

We want to compute $e_{k,r} * e_{l,s}$ and $e_{l,s} * e_{k,r}$ in the component $K^{G_{\mathbf{w}}\times {\mathbb{C}}}(Z({\mathbf{v}}^1,{\mathbf{v}}^3,{\mathbf{w}}))$.

Let us consider the intersection

$$\begin{equation*} p_{12}^{-1}{\mathfrak{P}}_k({\mathbf{v}}^2,{\mathbf{w}}) \cap p_{23}^{-1}{\mathfrak{P}}_{l}({\mathbf{v}}^3,{\mathbf{w}}) \qquad \text{(resp. $p_{12}^{-1}{\mathfrak{P}}_{l}({\mathbf{v}}^4,{\mathbf{w}}) \cap p_{23}^{-1}{\mathfrak{P}}_{k}({\mathbf{v}}^3,{\mathbf{w}})$)} \end{equation*}$$

in ${\mathfrak{M}}({\mathbf{v}}^1,{\mathbf{w}})\times {\mathfrak{M}}({\mathbf{v}}^2,{\mathbf{w}})\times {\mathfrak{M}}({\mathbf{v}}^3,{\mathbf{w}})$ (resp. ${\mathfrak{M}}({\mathbf{v}}^1,{\mathbf{w}})\times {\mathfrak{M}}({\mathbf{v}}^4,{\mathbf{w}})\times {\mathfrak{M}}({\mathbf{v}}^3,{\mathbf{w}})$).

Lemma 10.3.1.

The above two intersections are transversal respectively.

Proof.

The proof below is modeled on 45, 9.8, 9.9. We give the proof for $p_{12}^{-1}{\mathfrak{P}}_k({\mathbf{v}}^2,{\mathbf{w}}) \cap p_{23}^{-1}{\mathfrak{P}}_{l}({\mathbf{v}}^3,{\mathbf{w}})$. Then the same result for $p_{12}^{-1}{\mathfrak{P}}_{l}({\mathbf{v}}^4,{\mathbf{w}}) \cap p_{23}^{-1}{\mathfrak{P}}_{k}({\mathbf{v}}^3,{\mathbf{w}})$ follows by $k\leftrightarrow l$, ${\mathbf{v}}^2 \leftrightarrow {\mathbf{v}}^4$.

We consider the complex 5.1.1 for ${\mathfrak{P}}_k({\mathbf{v}}^2,{\mathbf{w}})$ and ${\mathfrak{P}}_l({\mathbf{v}}^3,{\mathbf{w}})$:

$$\begin{align*} & \operatorname {L}(V^1, V^2) \overset{\sigma ^{12}}{\longrightarrow } \operatorname {E}(V^1,V^2) \oplus \operatorname {L}(W, V^2) \oplus \operatorname {L}(V^1,W) \overset{\tau ^{12}}{\longrightarrow } \operatorname {L}(V^1, V^2) \oplus \mathcal{O}, \\ & \operatorname {L}(V^2, V^3) \overset{\sigma ^{23}}{\longrightarrow } \operatorname {E}(V^2,V^3) \oplus \operatorname {L}(W, V^3) \oplus \operatorname {L}(V^2,W) \overset{\tau ^{23}}{\longrightarrow } \operatorname {L}(V^2, V^3) \oplus \mathcal{O}, \end{align*}$$

where we use suffixes $12$, $23$ to distinguish endomorphisms. We have sections $s^{12}$ and $s^{23}$ of $\operatorname {Ker}\tau ^{12}/\operatorname {Im}\sigma ^{12}$ and $\operatorname {Ker}\tau ^{23}/\operatorname {Im}\sigma ^{23}$ respectively.

Identifying these vector bundles and sections with those of pull-backs to ${\mathfrak{M}}({\mathbf{v}}^1,{\mathbf{w}})\times {\mathfrak{M}}({\mathbf{v}}^2,{\mathbf{w}})\times {\mathfrak{M}}({\mathbf{v}}^3,{\mathbf{w}})$, we consider their zero loci $Z(s^{12}) = {\mathfrak{P}}_k({\mathbf{v}}^2,{\mathbf{w}})\times {\mathfrak{M}}({\mathbf{v}}^3,{\mathbf{w}})$ and $Z(s^{23}) = {\mathfrak{M}}({\mathbf{v}}^1,{\mathbf{w}})\times {\mathfrak{P}}_{l}({\mathbf{v}}^3,{\mathbf{w}})$.

As in the proof of 45, 5.7, we consider the transpose of $\nabla s^{12}$, $\nabla s^{23}$ via the symplectic form. Their sum gives a vector bundle endomorphism

$$\begin{equation*} \begin{split} {}^t (\nabla s^{12}) + {}^t (\nabla s^{22}) \colon & \operatorname {Ker}{}^t \sigma ^{12} / \operatorname {Im}{}^t \tau ^{12} \oplus \operatorname {Ker}{}^t \sigma ^{23} / \operatorname {Im}{}^t \tau ^{23} \\ &\qquad \to T{\mathfrak{M}}({\mathbf{v}}^1,{\mathbf{w}})\oplus T{\mathfrak{M}}({\mathbf{v}}^2,{\mathbf{w}})\oplus T{\mathfrak{M}}({\mathbf{v}}^3,{\mathbf{w}}). \end{split} \end{equation*}$$

It is enough to show that the kernel of ${}^t (\nabla s^{12}) + {}^t (\nabla s^{23})$ is zero at $(x^1, x^2, x^3)$.

Take representatives $(B^a , i^a , j^a )$ of $x^a$ ($a = 1,2,3$). Then we have $\xi ^{12}$, $\xi ^{23}$ which satisfy 5.1.3 for $(B^a , i^a , j^a)$. Suppose that

$$\begin{equation*} (C^{\prime 12}, a^{\prime 12}, b^{\prime 12}) \pmod {\operatorname {Im}{}^t \tau ^{12}} \oplus (C^{\prime 23}, a^{\prime 23}, b^{\prime 23}) \pmod {\operatorname {Im}{}^t \tau ^{23}} \end{equation*}$$

lies in the kernel. Then there exist $\gamma ^a\in \operatorname {L}(V^a, V^a)$ ($a = 1,2,3$) such that

$$\begin{equation} \begin{gathered} \left\{\begin{split} \varepsilon C^{\prime 12}\,\xi ^{12} &= \gamma ^1\, B^1 - B^1 \, \gamma ^1, \\ b^{\prime 12} &= \gamma ^1\, i^1, \\ -a^{\prime 12}\,\xi ^{12} &= - j^1 \,\gamma ^1, \end{split}\right. \qquad \left\{\begin{split} \varepsilon \,\xi ^{23} C^{\prime 23} &= \gamma ^3\, B^3 - B^3 \, \gamma ^3, \\ \xi ^{23}\,b^{\prime 23} &= \gamma ^3\, i^3, \\ -a^{\prime 23} &= - j^3 \,\gamma ^3, \end{split}\right. \\ \left\{\begin{split} \varepsilon (\xi ^{12} \, C^{\prime 12} + C^{\prime 23} \, \xi ^{23}) &= \gamma ^2\, B^2 - B^2 \, \gamma ^2,\\ \xi ^{12} \, b^{\prime 12} + b^{\prime 23} &= \gamma ^2\,i^2, \\ -a^{\prime 12} - a^{\prime 23}\xi ^{23} &= - j^2 \gamma ^2. \end{split}\right. \end{gathered} {\tag{10.3.2}} \end{equation}$$

Then we have

$$\begin{gather*} B^3\left( \xi ^{23}(\gamma ^2\xi ^{12}-\xi ^{12}\gamma ^1)-\gamma ^3\xi ^{23}\xi ^{12} \right) = \left( \xi ^{23}(\gamma ^2\xi ^{12}-\xi ^{12}\gamma ^1)-\gamma ^3\xi ^{23}\xi ^{12} \right) B^1, \\ j^3\left( \xi ^{23}(\gamma ^2\xi ^{12}-\xi ^{12}\gamma ^1)-\gamma ^3\xi ^{23}\xi ^{12} \right) = 0. \end{gather*}$$

Hence we have

$$\begin{equation} \xi ^{23}(\gamma ^2\xi ^{12}-\xi ^{12}\gamma ^1)-\gamma ^3\xi ^{23}\xi ^{12} = 0 {\tag{10.3.3}} \end{equation}$$

by the stability condition.

Consider the equation 10.3.3 at the vertex $l$. Since $k\neq l$, $\xi ^{12}_l$ is an isomorphism. Hence 10.3.3 implies that $\operatorname {Im}\xi ^{23}_l$ is invariant under $\gamma ^3_l$. Since $\operatorname {Im}\xi ^{23}_l$ is a codimension $1$ subspace, the induced map $\gamma ^3_l\colon V^3_l/\operatorname {Im}\xi ^{23}_l \to V^3_l/\operatorname {Im}\xi ^{23}_l$ is a scalar which we denote by $\lambda ^{23}$. Moreover, there exists a homomorphism $\zeta ^{23}_l\colon V^3_l \to V^2_l$ such that

$$\begin{equation*} \gamma ^3_l - \lambda ^{23}\operatorname {id}_{V^3_l} = \xi ^{23}_l \zeta ^{23}_l. \end{equation*}$$

For another vertex $l'\neq l$, $\xi ^{23}_{l'}$ is an isomorphism, hence we can define $\zeta ^{23}_{l'}$ so that the same equation holds also for the vertex $l'$. Thus we have

$$\begin{equation} \gamma ^3 - \lambda ^{23}\operatorname {id}_{V^3} = \xi ^{23} \zeta ^{23}. {\tag{10.3.4}} \end{equation}$$

Substituting 10.3.4 into 10.3.2 and using the injectivity of $\xi ^{23}$, we get

$$\begin{equation*} \left\{\begin{split} C^{\prime 23} &= \varepsilon (\zeta ^{23}B^3 - B^2 \,\zeta ^{23}), \\ a^{\prime 23} &= j^3 (\xi ^{23}\zeta ^{23} + \lambda ^{23}\,\operatorname {id}_{V^3}),\\ b^{\prime 23} &= (\zeta ^{23}\xi ^{23} + \lambda ^{23}\,\operatorname {id}_{V^2}) i^2. \end{split}\right. \end{equation*}$$

This means that $(C^{\prime 23}, a^{\prime 23}, b^{\prime 23}) = \,{}^{t}{\tau ^{23}}(\zeta ^{23}\oplus \lambda ^{23})$.

Substituting 10.3.4 into 10.3.3 and noticing $\xi ^{23}$ is injective, we obtain

$$\begin{equation} \left( \gamma ^2 - (\zeta ^{23}\xi ^{23} + \lambda ^{23}\operatorname {id}_{V^2}) \right)\xi ^{12} = \xi ^{12}\gamma ^1 . {\tag{10.3.5}} \end{equation}$$

Thus $\operatorname {Im}\xi ^{12}$ is invariant under $\gamma ^2 - (\zeta ^{23}\xi ^{23} + \lambda ^{23}\operatorname {id}_{V^2}).$ Arguing as above, we can find a constant $\lambda ^{12}$ and a homomorphism $\zeta ^{12}\colon V^2\to V^1$ such that

$$\begin{equation} \gamma ^2 - (\zeta ^{23}\xi ^{23} + \lambda ^{23}\operatorname {id}_{V^2}) - \lambda ^{12}\operatorname {id}_{V^2} = \xi ^{12}\zeta ^{12}. {\tag{10.3.6}} \end{equation}$$

Substituting this equation into 10.3.2, we get

$$\begin{equation*} \left\{\begin{split} C^{\prime 12} &= \varepsilon (\zeta ^{12}B^2 - B^1 \,\zeta ^{12}), \\ a^{\prime 12} &= j^2 (\xi ^{12}\zeta ^{12} + \lambda ^{12}\,\operatorname {id}_{V^2}),\\ b^{\prime 12} &= (\zeta ^{12}\xi ^{12} + \lambda ^{12}\,\operatorname {id}_{V^1}) i^1. \end{split}\right. \end{equation*}$$

This means that $(C^{\prime 12}, a^{\prime 12}, b^{\prime 12}) = \,{}^{t}{\tau ^{12}}(\zeta ^{12}\oplus \lambda ^{12})$. Hence ${}^t (\nabla s^{12}) + {}^t (\nabla s^{23})$ is injective.

■

Let us consider the variety $\widetilde{Z}_{kl}$ (resp. $\widetilde{Z}_{lk}$) of all pairs $(B,i,j)\in \mu ^{-1}(0)^{\operatorname {s}}$ and $S\subset V^3$ satisfying the following:

(a)

$S$ is a subspace with $\dim S = {\mathbf{v}}^1 = {\mathbf{v}}^3 - \alpha _k - \alpha _l$,

(b)

$S$ is $B$-stable,

(c)

$\operatorname {Im}i \subset S$,

(d)

the induced homomorphism $B_h\colon V_k^3/S_k \to V_l^3/S_l$ (resp. $B_{\overline{h}}\colon V_l^3/S_l \to V_k^3/S_k$) is zero for $h$ with $\operatorname {out}(h) = k$, $\operatorname {in}(h) = l$.

Then $\widetilde{Z}_{kl}/G_{{\mathbf{v}}^3}$ (resp. $\widetilde{Z}_{lk}/G_{{\mathbf{v}}^3}$) is isomorphic to $p_{12}^{-1}{\mathfrak{P}}_k({\mathbf{v}}^2,{\mathbf{w}}) \cap p_{23}^{-1}{\mathfrak{P}}_{l}({\mathbf{v}}^3,{\mathbf{w}})$ (resp. $p_{12}^{-1}{\mathfrak{P}}_{l}({\mathbf{v}}^2,{\mathbf{w}}) \cap p_{23}^{-1}{\mathfrak{P}}_{k}({\mathbf{v}}^3,{\mathbf{w}})$). The isomorphism is given by defining

$$\begin{align*} & (B^1,i^1,j^1) \overset{\operatorname {\scriptstyle def.}}{=}\text{the restriction of $(B,i,j)$ to $S$},\\ & (B^2,i^2,j^2) \overset{\operatorname {\scriptstyle def.}}{=}\text{the restriction of $(B,i,j)$ to $S'$},\\ \Big (\text{resp. } & (B^4,i^4,j^4) \overset{\operatorname {\scriptstyle def.}}{=}\text{the restriction of $(B,i,j)$ to $S''$}\Big ),\\ & (B^3,i^3,j^3) \overset{\operatorname {\scriptstyle def.}}{=}(B,i,j), \end{align*}$$

where $S'$ (resp. $S''$) is given by

$$\begin{equation*} S'_m \overset{\operatorname {\scriptstyle def.}}{=}\begin{cases} V_m &\text{if $m\neq l$}\\ S_l &\text{if $m = l$} \end{cases} \qquad \left(\text{resp. } S''_m \overset{\operatorname {\scriptstyle def.}}{=}\begin{cases} V_m &\text{if $m\neq k$}\\ S_k &\text{if $m = k$} \end{cases} \right). \end{equation*}$$

It is also clear that the restriction of $p_{13}$ to $\widetilde{Z}_{kl}/G_{{\mathbf{v}}^3}$ (resp. $\widetilde{Z}_{lk}/G_{{\mathbf{v}}^3}$) is an isomorphism onto its image. Hereafter, we identify $\widetilde{Z}_{kl}/G_{{\mathbf{v}}^3}$ (resp. $\widetilde{Z}_{lk}/G_{{\mathbf{v}}^3}$) with the image. Then $\widetilde{Z}_{kl}/G_{{\mathbf{v}}^3}$ and $\widetilde{Z}_{lk}/G_{{\mathbf{v}}^3}$ are closed subvarieties of $Z({\mathbf{v}}^1,{\mathbf{v}}^3;{\mathbf{w}})$. Let $i_{kl}\colon \widetilde{Z}_{kl}/G_{{\mathbf{v}}^3}\to Z({\mathbf{v}}^1,{\mathbf{v}}^3;{\mathbf{w}})$ (resp. $i_{lk}\colon \widetilde{Z}_{lk}/G_{{\mathbf{v}}^3}\to Z({\mathbf{v}}^1,{\mathbf{v}}^3;{\mathbf{w}})$) denote the inclusion.

The quotient $V_k^3/S_k$ (resp. $V_l^3/S_l$) forms a line bundle over

$$\begin{equation*} \widetilde{Z}_{kl}/G_{{\mathbf{v}}^3} = p_{12}^{-1}{\mathfrak{P}}_k({\mathbf{v}}^2,{\mathbf{w}}) \cap p_{23}^{-1}{\mathfrak{P}}_{l}({\mathbf{v}}^3,{\mathbf{w}}) \end{equation*}$$

(resp. $\widetilde{Z}_{lk}/G_{{\mathbf{v}}^3} = p_{12}^{-1}{\mathfrak{P}}_{l}({\mathbf{v}}^2,{\mathbf{w}}) \cap p_{23}^{-1}{\mathfrak{P}}_{k}({\mathbf{v}}^3,{\mathbf{w}})).$ By the above consideration, $e_{k,r}\ast e_{l,s}$ (resp. $e_{l,s}\ast e_{k,r}$) is represented by

$$\begin{multline} (-1)^{\langle h_k, {\mathbf{C}}_{\overline{\Omega }}{\mathbf{v}}^2\rangle +\langle h_l, {\mathbf{C}}_{\overline{\Omega }}{\mathbf{v}}^3\rangle } i_{kl*}[ (q^{-1}V^3_k/S_k)^{\otimes r-\langle h_k, {\mathbf{C}}''{\mathbf{v}}^2\rangle } \otimes (q^{-1}V^3_l/S_l)^{\otimes s-\langle h_l, {\mathbf{C}}''{\mathbf{v}}^3\rangle }\\ \otimes \det C_k^{\prime \bullet }({\mathbf{v}}^2,{\mathbf{w}})^* \otimes \det C_l^{\prime \bullet }({\mathbf{v}}^3,{\mathbf{w}})^*] {\tag{10.3.7}} \end{multline}$$

(resp.

$$\begin{multline*} (-1)^{\langle h_l, {\mathbf{C}}_{\overline{\Omega }}{\mathbf{v}}^4\rangle +\langle h_k, {\mathbf{C}}_{\overline{\Omega }}{\mathbf{v}}^3\rangle } i_{lk*}[ (q^{-1}V^3_l/S_l)^{\otimes s-\langle h_l, {\mathbf{C}}''{\mathbf{v}}^4\rangle } \otimes (q^{-1}V^3_k/S_k)^{\otimes r-\langle h_k, {\mathbf{C}}''{\mathbf{v}}^3\rangle }\\ \otimes \det C_l^{\prime \bullet }({\mathbf{v}}^4,{\mathbf{w}})^* \otimes \det C_k^{\prime \bullet }({\mathbf{v}}^3,{\mathbf{w}})^*]). \end{multline*}$$

Note that we have

$$\begin{equation} \begin{gathered} \langle h_k, {\mathbf{C}}_{\overline{\Omega }}{\mathbf{v}}^2\rangle +\langle h_l, {\mathbf{C}}_{\overline{\Omega }}{\mathbf{v}}^3\rangle = \langle h_l, {\mathbf{C}}_{\overline{\Omega }}{\mathbf{v}}^4\rangle +\langle h_k, {\mathbf{C}}_{\overline{\Omega }}{\mathbf{v}}^3\rangle \pm (\alpha _k,\alpha _l),\\ \langle h_k, {\mathbf{C}}''{\mathbf{v}}^2\rangle = \langle h_k, {\mathbf{C}}''{\mathbf{v}}^3\rangle , \qquad \langle h_l, {\mathbf{C}}''{\mathbf{v}}^3\rangle = \langle h_l, {\mathbf{C}}''{\mathbf{v}}^4\rangle , \\ \det C_k^{\prime \bullet }({\mathbf{v}}^2,{\mathbf{w}}) = \det C_k^{\prime \bullet }({\mathbf{v}}^3,{\mathbf{w}})\otimes (q^{-1}V^3_l/S_l)^{\otimes (\alpha _k,\alpha _l)},\\ \det C_l^{\prime \bullet }({\mathbf{v}}^4,{\mathbf{w}}) = \det C_l^{\prime \bullet }({\mathbf{v}}^3,{\mathbf{w}})\otimes (q^{-1}V^3_k/S_k)^{\otimes (\alpha _k,\alpha _l)}. \end{gathered} {\tag{10.3.8}} \end{equation}$$

Set $b' = -(\alpha _k,\alpha _l)$. We consider

$$\begin{equation*} \bigoplus _{\operatorname {out}(h)=k, \operatorname {in}(h)=l} B_{\overline{h}} \end{equation*}$$

as a section of the vector bundle $q [b']_q \operatorname {Hom}(V^3_l/S_l, V^3_k/S_k)$ over $\widetilde{Z}_{kl}/G_{{\mathbf{v}}^3}$. Let us denote it by $s_{kl}$. Similarly $\bigoplus _{\operatorname {out}(h)=k, \operatorname {in}(h)=l} B_{h}$ is a section (denoted by $s_{lk}$) of the vector bundle $q [b']_q \operatorname {Hom}(V^3_k/S_k, V^3_l/S_l)$ over $\widetilde{Z}_{lk}/G_{{\mathbf{v}}^3}$.

Lemma 10.3.9.

The section $s_{kl}$ (resp. $s_{lk}$) is transversal to the zero section (if it vanishes somewhere).

Proof.

Fix a subspace $S\subset V^3$ with $\dim S = {\mathbf{v}}^1$. Let $P$ be the parabolic subgroup of $G_{{\mathbf{v}}^3}$ consisting of elements which preserve $S$. We also fix a complementary subspace $T$. Thus we have $V^3 = S\oplus T$. We will check the assertion for $s_{kl}$. The assertion for $s_{lk}$ follows if we exchange $k$ and $l$.

We consider

$$\begin{equation*} \widetilde{{\mathbf{M}}}\overset{\operatorname {\scriptstyle def.}}{=}\left\{ (B,i,j)\in {\mathbf{M}}({\mathbf{v}}^3,{\mathbf{w}})\left| \begin{aligned} & B(S) \subset S,\; \operatorname {Im}i\subset S, \\ & \text{$B_h\colon V^3_k/S_k \to V^3_l/S_l$ is $0$ for $h$} \\ &\qquad \qquad \qquad \qquad \text{ with $\operatorname {out}(h) = k$, $\operatorname {in}(h) = l$} \end{aligned} \right\}\right.. \end{equation*}$$

It is a linear subspace of ${\mathbf{M}}({\mathbf{v}}^3,{\mathbf{w}})$. Let $\widetilde{\mu }\colon \widetilde{{\mathbf{M}}}\to \operatorname {L}(V^3,S)$ be the composition of the restriction of the moment map $\mu \colon {\mathbf{M}}({\mathbf{v}}^3,{\mathbf{w}})\to \operatorname {L}(V^3,V^3)$ to $\widetilde{{\mathbf{M}}}$ and the projection $\operatorname {L}(V^3,V^3)\to \operatorname {L}(V^3,S)$. Let $\widetilde{\mu }^{-1}(0)^{\operatorname {s}}$ denote the set of $(B,i,j)\in \widetilde{\mu }^{-1}(0)$ which is stable. It is preserved under the action of $P$ and we have a $G_{{\mathbf{v}}^3}$-equivariant isomorphism

$$\begin{equation*} G_{{\mathbf{v}}^3}\times _{P} \widetilde{\mu }^{-1}(0)^{\operatorname {s}} \cong \widetilde{Z}_{kl}. \end{equation*}$$

Note that the $\operatorname {L}(V^3,T)$-part of the moment map $\mu$ vanishes on $\widetilde{{\mathbf{M}}}$ thanks to the definition of $\widetilde{{\mathbf{M}}}$.

The assertion follows if we check that

$$\begin{equation*} d\widetilde{\mu }\oplus \Pi \colon \widetilde{{\mathbf{M}}}\to \operatorname {L}(V^3,S) \oplus q [b']_q \operatorname {Hom}(V^3_l/S_l, V^3_k/S_k) \end{equation*}$$

is surjective at $(B,i,j)\in \widetilde{\mu }^{-1}(0)^{\operatorname {s}}$. Here $\Pi \colon \widetilde{{\mathbf{M}}}\to q [b']_q \operatorname {Hom}(V^3_l/S_l, V^3_k/S_k)$ is the natural projection. Thus it is enough to show that $d\widetilde{\mu }\colon \widetilde{{\mathbf{M}}}\cap \operatorname {Ker}\Pi \to \operatorname {L}(V^3,S)$ is surjective.

Suppose that $\zeta \in \operatorname {L}(S,V^3)$ is orthogonal to $d\widetilde{\mu }(\widetilde{{\mathbf{M}}}\cap \operatorname {Ker}\Pi )$, namely

$$\begin{equation*} \operatorname {tr}\left(\varepsilon (\delta B\,B + B\,\delta B)\zeta + \delta i\,j\,\zeta + i\,\delta j\,\zeta \right) = 0 \end{equation*}$$

for any $(\delta B, \delta i, \delta j)\in \widetilde{{\mathbf{M}}}\cap \operatorname {Ker}\Pi$. Hence we have

$$\begin{equation*} 0 = j\zeta \in \operatorname {L}(S,W), \qquad 0 = B\zeta - \zeta B|_S \in \operatorname {L}(S,V^3), \end{equation*}$$

where $B|_S$ is the restriction of $B$ to $S$. Therefore the image of $\zeta$ is invariant under $B$ and contained in $\operatorname {Ker}j$. By the stability condition, we have $\zeta = 0$. Thus we have proved the assertion.

■

Let $Z(s_{kl})$ (resp. $Z(s_{lk})$) be the zero locus of $s_{kl}$ (resp. $s_{lk}$). By Lemma 10.3.9, we have the following exact sequence (Koszul complex) on $\widetilde{Z}_{kl}/G_{{\mathbf{v}}^3}$ (resp. $\widetilde{Z}_{lk}/G_{{\mathbf{v}}^3}$):

$$\begin{multline*} 0 \to {\textstyle \bigwedge }^{\max } (q [b']_q \operatorname {Hom}(V^3_l/S_l, V^3_k/S_k))^* \to \cdots \to (q [b']_q \operatorname {Hom}(V^3_l/S_l, V^3_k/S_k))^* \\ \to \mathcal{O}_{\widetilde{Z}_{kl}/G_{{\mathbf{v}}^3}} \to \mathcal{O}_{Z(s_{kl})} \to 0 \end{multline*}$$

(resp.

$$\begin{multline*} 0 \to {\textstyle \bigwedge }^{\max } (q [b']_q \operatorname {Hom}(V^3_k/S_k, V^3_l/S_l))^* \to \cdots \to (q [b']_q \operatorname {Hom}(V^3_k/S_k, V^3_l/S_l))^* \\ \to \mathcal{O}_{\widetilde{Z}_{lk}/G_{{\mathbf{v}}^3}} \to \mathcal{O}_{Z(s_{lk})} \to 0). \end{multline*}$$

Hence we have the following equality in $K^{G_{{\mathbf{w}}}\times {\mathbb{C}}^*} (\widetilde{Z}_{kl}/G_{{\mathbf{v}}^3})$ (resp. $K^{G_{{\mathbf{w}}}\times {\mathbb{C}}^*} (\widetilde{Z}_{lk}/G_{{\mathbf{v}}^3})$):

$$\begin{align*} & \mathcal{O}_{Z(s_{kl})} = {\textstyle \bigwedge }_{-1} (q [b']_q \operatorname {Hom}(V^3_l/S_l, V^3_k/S_k))^* \\ \Big ( \text{resp. } & \mathcal{O}_{Z(s_{lk})} = {\textstyle \bigwedge }_{-1} (q [b']_q \operatorname {Hom}(V^3_k/S_k, V^3_l/S_l))^* \Big ). \end{align*}$$

Both $Z(s_{kl})$ and $Z(s_{lk})$ consist of all pairs $(B,i,j)\in \mu ^{-1}(0)^{\operatorname {s}}$ and $S\subset V^3$ satisfying the following:

(a)

$S$ is a subspace with $\dim S = {\mathbf{v}}^1 = {\mathbf{v}}^3 - \alpha _k - \alpha _l$,

(b)

$S$ is $B$-stable,

(c)

$\operatorname {Im}i \subset S$,

(d)

the induced homomorphism $B\colon V/S \to V/S$ is zero,

modulo the action of $G_{{\mathbf{v}}^3}$. In particular, we have $Z(s_{kl}) = Z(s_{lk})$. Hence we have

$$\begin{equation*} i_{kl*}\left[ {\textstyle \bigwedge }_{-1} (q [b']_q \operatorname {Hom}(V^3_l/S_l, V^3_k/S_k))^*\right] = i_{lk*}\left[ {\textstyle \bigwedge }_{-1} (q [b']_q \operatorname {Hom}(V^3_k/S_k, V^3_l/S_l))^*\right]. \end{equation*}$$

This implies

$$\begin{equation} \begin{split} & i_{kl*}\left[(q^{-1}V^3_k/S_k)^{\otimes r} \otimes (q^{-1} V^3_l/S_l)^{\otimes s} \otimes {\textstyle \bigwedge }_{-1} (q [b']_q \operatorname {Hom}(V^3_l/S_l, V^3_k/S_k))^*\right] \\ &\quad = i_{lk*}\left[(q^{-1}V^3_k/S_k)^{\otimes r} \otimes (q^{-1}V^3_l/S_l)^{\otimes s} \otimes {\textstyle \bigwedge }_{-1} (q [b']_q \operatorname {Hom}(V^3_k/S_k, V^3_l/S_l))^*\right] \end{split} \tag{10.3.10} \end{equation}$$

by the projection formula 6.5.1. Multiplying this equality by $z^{-r} w^{-s}$ and taking the sum with respect to $r$ and $s$, we get

$$\begin{align*} & \prod _{p=1}^{b'} (1 - q^{b'-2p}\frac{w}{z}) \sum _{r,s=-\infty }^\infty i_{kl*}\left[(q^{-1}V^3_k/S_k)^{\otimes r} \otimes (q^{-1} V^3_l/S_l)^{\otimes s}\right] z^{-r} w^{-s}\\ &\qquad = \prod _{p=1}^{b'} (1 - q^{b'-2p}\frac{z}{w}) \sum _{r,s=-\infty }^\infty i_{lk*}\left[(q^{-1}V^3_k/S_k)^{\otimes r}\otimes (q^{-1} V^3_l/S_l)^{\otimes s}\right] z^{-r} w^{-s}. \end{align*}$$

Comparing this with 10.3.7 and using 10.3.8, we get 1.2.10.

10.4. Relation 1.2.11

We give the proof of 1.2.11 for $\pm = +$ in this subsection, assuming other relations. (The relations 1.2.8 with $k = l$ and 1.2.9 will be checked in the next section, but its proof is independent of results in this subsection.) The relation 1.2.11 for $\pm = -$ can be proved in a similar way, and hence is omitted.

By the proof of 45, 9.3, operators $e_{k,0}$, $f_{k,0}$ acting on $K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(Z({\mathbf{w}}))$ are locally nilpotent. (See also Lemma 13.2.4 below.) It is known that the constant term of 1.2.11, i.e.,

$$\begin{equation} \sum _{p=0}^{b}(-1)^p \begin{bmatrix} b \\ p\end{bmatrix}_{q_k} e_{k,0}^p e_{l,0} e_{k,0}^{b-p} = 0, {\tag{10.4.1}} \end{equation}$$

can be deduced from the other relations and the local nilpotency of $e_{k,0}$, $f_{k,0}$ (see, e.g., 13, 4.3.2 for the proof for $q = 1$). Thus our task is to reduce

$$\begin{equation} \sum _{\sigma \in S_b} \sum _{p=0}^{b}(-1)^p \begin{bmatrix} b \\ p\end{bmatrix}_{q_k} e_{k,r_{\sigma (1)}}\cdots e_{k, r_{\sigma (p)}} e_{l,s} e_{k, r_{\sigma (p+1)}}\cdots e_{k,r_{\sigma (b)}} = 0 {\tag{10.4.2}} \end{equation}$$

to 10.4.1. This reduction was done by Grojnowski 23, but we reproduce it here for the sake of completeness.

For $p\in \{0,1,\dots , b\}$, let ${\mathbf{v}}^0,\dots ,{\mathbf{v}}^{b+1}$ be dimension vectors with

$$\begin{equation*} {\mathbf{v}}^i = \begin{cases} {\mathbf{v}}^{i-1} + \alpha _k & \text{if $i\neq p+1$}, \\ {\mathbf{v}}^{i-1} + \alpha _l & \text{if $i = p+1$}. \end{cases} \end{equation*}$$

Let

$$\begin{equation*} \pi _{ij}\colon {\mathfrak{M}}({\mathbf{v}}^0,{\mathbf{w}})\times \cdots \times {\mathfrak{M}}({\mathbf{v}}^{b+1},{\mathbf{w}}) \to {\mathfrak{M}}({\mathbf{v}}^{i},{\mathbf{w}})\times {\mathfrak{M}}({\mathbf{v}}^j,{\mathbf{w}}) \end{equation*}$$

be the projection. Let

$$\begin{equation} \begin{split} & \widehat{{\mathfrak{P}}}_p \overset{\operatorname {\scriptstyle def.}}{=}\pi _{12}^{-1}({\mathfrak{P}}_k({\mathbf{v}}^1,{\mathbf{w}}))\cap \cdots \cap \pi _{p-1,p}^{-1}({\mathfrak{P}}_k({\mathbf{v}}^p,{\mathbf{w}})) \\ &\qquad \cap \pi _{p,p+1}^{-1}({\mathfrak{P}}_l({\mathbf{v}}^{p+1},{\mathbf{w}}))\cap \pi _{p+1,p+2}^{-1}({\mathfrak{P}}_k({\mathbf{v}}^{p+2},{\mathbf{w}}))\cap \cdots \cap \pi _{b,b+1}^{-1}({\mathfrak{P}}_k({\mathbf{v}}^{b+1},{\mathbf{w}})). \end{split}{\tag{10.4.3}} \end{equation}$$

This is equal to

$$\begin{equation*} \{ (B,i,j,V^0\subset \cdots \subset V^{b+1}) \mid \text{as below}\} /G_{{\mathbf{v}}^{b+1}}, \end{equation*}$$

(a)

$(B,i,j)\in \mu ^{-1}(0)^{\operatorname {s}}$ (in ${\mathbf{M}}({\mathbf{v}}^{b+1},{\mathbf{w}})$),

(b)

$V^i$ is a $B$-invariant subspace containing the image of $i$ with $\dim V^i = {\mathbf{v}}^i$.

In particular, we have line bundles $V^i/V^{i-1}$ on $\widehat{{\mathfrak{P}}}_p$ ($i = 1$, …, $b+1$). By the definition, there exists a line bundle $\mathfrak{L}_p(r_1,\dots ,r_b;s)\in K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(\widehat{{\mathfrak{P}}}_p)$ such that

$$\begin{equation*} e_{k,r_1}\cdots e_{k,r_p} e_{l,s} e_{k,r_{p+1}}\cdots e_{k,r_b} = \pi _{0,b+1 *} \mathfrak{L}_p(r_1,\dots ,r_b;s). \end{equation*}$$

Moreover, we have

$$\begin{multline} \mathfrak{L}_p(r_1,\dots ,r_b;s) = \left(q^{-1} V^1_k/V^0_k \right)^{\otimes r_1} \otimes \cdots \otimes \left(q^{-1} V^p_k/V^{p-1}_k \right)^{\otimes r_p} \otimes \left(q^{-1} V^{p+1}_l/V^{p}_l \right)^{\otimes s} \\ \otimes \left(q^{-1} V^{p+2}_k/V^{p+1}_k \right)^{\otimes r_{p+1}} \otimes \cdots \otimes \left(q^{-1} V^{b+1}_k/V^{b}_k \right)^{\otimes r_{b}} \otimes \mathfrak{L}_p(0,\dots ,0;0). {\tag{10.4.4}} \end{multline}$$

Now consider the symmetrization. By 10.4.4, we have

$$\begin{equation*} \sum _{\sigma \in S_b} \mathfrak{L}_p(r_{\sigma (1)},\dots ,r_{\sigma (b)};s) = T(V_k^{b+1}/V_k^0)\otimes \left(q^{-1} V^{p+1}_l/V^{p}_l \right)^{\otimes s} \otimes \mathfrak{L}_p(0,\dots ,0;0) \end{equation*}$$

for some tensor product $T(V_k^{b+1}/V_k^0)$ of exterior products of the bundle $V_k^{b+1}/V_k^0$ and its dual. (In the notation in §11.4 below, $T(V_k^{b+1}/V_k^0)$ corresponds to the symmetric function $\sum _{\sigma \in S_b} x_1^{r_{\sigma (1)}}\cdots x_b^{r_{\sigma (b)}}$.) Note that we have $q^{-1} V^{p+1}_l/V^{p}_l = q^{-1} V^{b+1}_l/V^{0}_l$. Thus $T(V_k^{b+1}/V_k^0)\otimes \left(q^{-1} V^{b+1}_l/V^{0}_l \right)^{\otimes s}$ can be considered as a vector bundle over $\pi _{0,b+1}(\widehat{{\mathfrak{P}}}_p)$. Then the projection formula implies that

$$\begin{align*} & \sum _{\sigma \in S_b} \pi _{0,b+1*} \mathfrak{L}_p(r_{\sigma (1)},\dots ,r_{\sigma (b)};s)\\ &\qquad = T(V_k^{b+1}/V_k^0)\otimes \left(q^{-1} V^{b+1}_l/V^{0}_l \right)^{\otimes s} \otimes \pi _{0,b+1*}\mathfrak{L}_p(0,\dots ,0;0). \end{align*}$$

Noticing that $T(V_k^{b+1}/V_k^0)\otimes \left(q^{-1} V^{b+1}_l/V^{0}_l \right)^{\otimes s}$ is independent of $p$, we can derive 10.4.2 from 10.4.1.

11. Relations (II)

The purpose of this section is to check the relations 1.2.8 with $k = l$ and 1.2.9. Our strategy is the following. We first reduce the computation of the convolution product to the case of the graph of type $A_1$ using results in §8 and introducing modifications of quiver varieties and Hecke correspondences. Then we perform the computation using the explicit description of the equivariant $K$-theory for quiver varieties for the graph of type $A_1$.

In this section we fix a vertex $k$.

11.1. Modifications of quiver varieties

We take a collection of vector spaces $V = (V_l)_{l\in I}$ with $\dim V = {\mathbf{v}}$. Let $\mu _k$ be the $\operatorname {Hom}(V_k,V_k)$-component of $\mu \colon {\mathbf{M}}({\mathbf{v}},{\mathbf{w}})\to \operatorname {L}(V,V)$, i.e.,

$$\begin{equation*} \mu _k(B,i,j) \overset{\operatorname {\scriptstyle def.}}{=}\sum _{\operatorname {in}(h)=k} \varepsilon (h) B_h B_{\overline{h}} + i_k j_k. \end{equation*}$$

Let

$$\begin{multline*} \widetilde{{\mathfrak{M}}}({\mathbf{v}},{\mathbf{w}}) \overset{\operatorname {\scriptstyle def.}}{=}\Big \{ (B,i,j)\in \mu _k^{-1}(0) \Big | \text{$\bigoplus _{\operatorname {out}(h)=k} B_h \oplus j_k\colon V_k$}\\ \text{ $\to \bigoplus _{\operatorname {out}(h)=k} V_{\operatorname {in}(h)}\oplus W_k$ is injective} \Big \}\Big / \operatorname {GL}(V_k). \end{multline*}$$

This is a product of the quiver variety for the graph of type $A_1$ and the affine space:

$$\begin{equation*} \widetilde{{\mathfrak{M}}}({\mathbf{v}},{\mathbf{w}}) = {\mathfrak{M}}(v_k,N) \times {\mathbf{M}}'({\mathbf{v}},{\mathbf{w}}), \end{equation*}$$

where

$$\begin{gather*} v_k = \dim V_k, \qquad N = - \sum _{l:l\neq k} (\alpha _k,\alpha _l)\dim V_l+\dim W_k, \\ {\mathbf{M}}'({\mathbf{v}},{\mathbf{w}}) \overset{\operatorname {\scriptstyle def.}}{=}\bigoplus _{h\colon \substack{\operatorname {in}(h)\neq k\\ \operatorname {out}(h)\neq k}} \operatorname {Hom}(V_{\operatorname {out}(h)}, V_{\operatorname {in}(h)}) \oplus \bigoplus _{l\colon l\neq k} \operatorname {Hom}(W_l, V_l)\oplus \operatorname {Hom}(V_l,W_l). \end{gather*}$$

Moreover, the variety ${\mathfrak{M}}(v_k,N)$ is isomorphic to the cotangent bundle of the Grassmann manifold $G(v_k,N)$ of $v_k$-dimensional subspaces in the $N$-dimensional space. (See 44, Chap. 7 for details.) The isomorphism is given as follows: ${\mathfrak{M}}(v_k,N)$ is the set of $\operatorname {GL}(v_k,{\mathbb{C}})$-orbits of $i\colon {\mathbb{C}}^N \to {\mathbb{C}}^{v_k}$, $j\colon {\mathbb{C}}^{v_k}\to {\mathbb{C}}^N$ such that

(a)

$ij = 0$,

(b)

$j$ is injective.

The action is given by $(i,j)\mapsto (gi, jg^{-1})$. Then

$$\begin{equation*} {\mathfrak{M}}(v_k,N)\ni G\cdot (i,j) \mapsto (\operatorname {Image}j\subset {\mathbb{C}}^N) \end{equation*}$$

defines a map ${\mathfrak{M}}(v_k,N)\to G(v_k,N)$, and the linear map

$$\begin{equation*} ji\colon {\mathbb{C}}^N/\operatorname {Image}j \longrightarrow \operatorname {Image}j \end{equation*}$$

defines a cotangent vector at $\operatorname {Image}j$.

Let $\mu ^{-1}(0)^{\operatorname {s}}$ be as in Definition 2.3.1 and let $\mu _k^{-1}(0)^{\operatorname {s}}$ be the set of stable points in $\mu _k^{-1}(0)$. Although the stability condition 2.3.1 was defined only for $(B,i,j)\in \mu ^{-1}(0)$, it can be defined for any $(B,i,j)\in {\mathbf{M}}({\mathbf{v}},{\mathbf{w}})$. Let

$$\begin{equation*} \widetilde{{\mathfrak{M}}}^{\circ }({\mathbf{v}},{\mathbf{w}}) \overset{\operatorname {\scriptstyle def.}}{=}\mu _k^{-1}(0)^{\operatorname {s}}/\operatorname {GL}(V_k), \qquad \widehat{{\mathfrak{M}}}({\mathbf{v}},{\mathbf{w}}) \overset{\operatorname {\scriptstyle def.}}{=}\mu ^{-1}(0)^{\operatorname {s}}/\operatorname {GL}(V_k). \end{equation*}$$

We have a natural action of

$$\begin{equation*} G_{\mathbf{v}}' \overset{\operatorname {\scriptstyle def.}}{=}\prod _{l:l \neq k} \operatorname {GL}(V_l) \end{equation*}$$

on $\widetilde{{\mathfrak{M}}}({\mathbf{v}},{\mathbf{w}})$, $\widetilde{{\mathfrak{M}}}^\circ ({\mathbf{v}},{\mathbf{w}})$ and $\widehat{{\mathfrak{M}}}({\mathbf{v}},{\mathbf{w}})$. We have the following relations between these varieties:

(a)

$\widetilde{{\mathfrak{M}}}^{\circ }({\mathbf{v}},{\mathbf{w}})$ is an open subvariety of $\widetilde{{\mathfrak{M}}}({\mathbf{v}},{\mathbf{w}})$,

(b)

$\widehat{{\mathfrak{M}}}({\mathbf{v}},{\mathbf{w}})$ is a nonsingular closed subvariety of $\widetilde{{\mathfrak{M}}}^\circ ({\mathbf{v}},{\mathbf{w}})$ (defined by the equation $\mu _l = 0$ for $l\neq k$),

(c)

$\widehat{{\mathfrak{M}}}({\mathbf{v}},{\mathbf{w}})$ is a principal $G_{\mathbf{v}}'$-bundle over ${\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})$.

The vector space $V_k$ defines vector bundles $\widetilde{{\mathfrak{M}}}({\mathbf{v}},{\mathbf{w}})$, $\widetilde{{\mathfrak{M}}}^\circ ({\mathbf{v}},{\mathbf{w}})$, $\widehat{{\mathfrak{M}}}({\mathbf{v}},{\mathbf{w}})$ and ${\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})$. We denote all of them by $V_k$ for brevity, hoping that it causes no confusion.

11.2. Modifications of Hecke correspondences

Fix $n \in {\mathbb{Z}}_{> 0}$. Take collections of vector spaces $V^1 = (V^1_l)_{l\in I}$, $V^2 = (V^2_l)_{l\in I}$ whose dimension vectors ${\mathbf{v}}^1$, ${\mathbf{v}}^2$ satisfy ${\mathbf{v}}^2 = {\mathbf{v}}^1 + n\alpha _k$. (For the proof of Theorem 9.4.1, it is enough to consider the case $n = 1$. But we study general $n$ for a later purpose.) These data will be fixed throughout this subsection, and we use the following notation:

$$\begin{gather*} \widetilde{{\mathfrak{M}}}_1 = \widetilde{{\mathfrak{M}}}({\mathbf{v}}^1,{\mathbf{w}}),\; \widetilde{{\mathfrak{M}}}_2 = \widetilde{{\mathfrak{M}}}({\mathbf{v}}^2,{\mathbf{w}}),\; \widetilde{{\mathfrak{M}}}_1^\circ = \widetilde{{\mathfrak{M}}}^\circ ({\mathbf{v}}^1,{\mathbf{w}}),\; \widetilde{{\mathfrak{M}}}_2^\circ = \widetilde{{\mathfrak{M}}}^\circ ({\mathbf{v}}^2,{\mathbf{w}}),\; \\ \widehat{{\mathfrak{M}}}_1 = \widehat{{\mathfrak{M}}}({\mathbf{v}}^1,{\mathbf{w}}),\; \widehat{{\mathfrak{M}}}_2 = \widehat{{\mathfrak{M}}}({\mathbf{v}}^2,{\mathbf{w}}),\; {\mathfrak{M}}_1 = {\mathfrak{M}}({\mathbf{v}}^1,{\mathbf{w}}), \; {\mathfrak{M}}_2 = {\mathfrak{M}}({\mathbf{v}}^2,{\mathbf{w}}). \end{gather*}$$

Consider $\widetilde{{\mathfrak{M}}}_1$ and $\widetilde{{\mathfrak{M}}}_2$. These varieties are products of quiver varieties for the graph of type $A_1$ and the affine space. We fix an isomorphism $V^1_l \cong V^2_l$ for $l\neq k$. Then we have identifications $G'_{{\mathbf{v}}^1}\cong G'_{{\mathbf{v}}^2}$ and ${\mathbf{M}}'({\mathbf{v}}^1,{\mathbf{w}})\cong {\mathbf{M}}'({\mathbf{v}}^2,{\mathbf{w}})$. We write them as $G'$ and ${\mathbf{M}}'$ respectively for brevity. We write $V_l$ for $V^1_l$ and $V^2_l$ for $l\neq k$. Let us define a subvariety $\widetilde{{\mathfrak{P}}}_k^{(n)}\subset \widetilde{{\mathfrak{M}}}_1\times \widetilde{{\mathfrak{M}}}_2$ as the product of the Hecke correspondence for the graph of type $A_1$ and the diagonal for the affine space. Namely

$$\begin{equation*} \begin{matrix} & \widetilde{{\mathfrak{P}}}_{k}^{(n)} & \overset{\operatorname {\scriptstyle def.}}{=}& {\mathfrak{P}}_{k}^{(n)}(v^2_k,N)\times \Delta {\mathbf{M}}' \\ & \cap & & \cap \\ & \widetilde{{\mathfrak{M}}}_1\times \widetilde{{\mathfrak{M}}}_2 & = & {\mathfrak{M}}(v^2_k-n,N)\times {\mathfrak{M}}(v^2_k,N)\times {\mathbf{M}}'\times {\mathbf{M}}', \end{matrix} \end{equation*}$$

where $v_k^2 = \dim V_k^2$ and

$$\begin{equation*} N = - \sum _{l:l\neq k} (\alpha _k,\alpha _l)\dim V^1_l+\dim W_k = - \sum _{l:l\neq k} (\alpha _k,\alpha _l)\dim V^2_l+\dim W_k. \end{equation*}$$

Here ${\mathfrak{P}}_{k}^{(n)}(v^2_k,N)\subset {\mathfrak{M}}(v^2_k-n,N)\times {\mathfrak{M}}(v^2_k,N)$ is the generalization of the Hecke correspondence introduced in 5.3.1. Since the graph is of type $A_1$, it is isomorphic to the conormal bundle of

$$\begin{equation*} O^{(n)}(v_k^2,N) \overset{\operatorname {\scriptstyle def.}}{=}\{ (V^1_k, V^2_k) \in G(v^2_k-n, N)\times G(v^2_k, N) \mid V^1_k \subset V^2_k \}. \end{equation*}$$

The quotient $V^2_k/V^1_k$ defines a vector bundle over ${\mathfrak{P}}_{k}^{(n)}(v^2_k,N)$ of rank $n$.

We have

$$\begin{equation} \widetilde{{\mathfrak{P}}}_{k}^{(n)}\cap \left(\widetilde{{\mathfrak{M}}}^\circ _1\times \widetilde{{\mathfrak{M}}}_2\right) \subset \widetilde{{\mathfrak{M}}}^\circ _1\times \widetilde{{\mathfrak{M}}}^\circ _2, \qquad \widetilde{{\mathfrak{P}}}_{k}^{(n)}\cap \left(\widetilde{{\mathfrak{M}}}_1\times \widetilde{{\mathfrak{M}}}^\circ _2\right) \subset \widetilde{{\mathfrak{M}}}^\circ _1\times \widetilde{{\mathfrak{M}}}^\circ _2. {\tag{11.2.1}} \end{equation}$$

The latter inclusion is obvious from the definition of stability, and the former one follows from the argument in 45, Proof of 4.5. Let

$$\begin{equation*} \widetilde{{\mathfrak{P}}}_{k}^{(n)\circ } \overset{\operatorname {\scriptstyle def.}}{=}\widetilde{{\mathfrak{P}}}_{k}^{(n)}\cap \left(\widetilde{{\mathfrak{M}}}^\circ _1\times \widetilde{{\mathfrak{M}}}^\circ _2\right). \end{equation*}$$

For $([B^1,i^1,j^1], [B^2,i^2,j^2]) \in \widetilde{{\mathfrak{P}}}_{k}^{(n)}$, the first factor $[B^1,i^1,j^1]$ satisfies $\mu (B^1,i^1,j^1) = 0$ if and only if the other factor $[B^2,i^2,j^2]$ also satisfies $\mu (B^2,i^2,j^2) = 0$. This implies that

$$\begin{equation} \widetilde{{\mathfrak{P}}}_{k}^{(n)\circ }\cap \left(\widehat{{\mathfrak{M}}}_1\times \widetilde{{\mathfrak{M}}}^\circ _2\right) \subset \widehat{{\mathfrak{M}}}_1\times \widehat{{\mathfrak{M}}}_2, \qquad \widetilde{{\mathfrak{P}}}_{k}^{(n)\circ }\cap \left(\widetilde{{\mathfrak{M}}}^\circ _1\times \widehat{{\mathfrak{M}}}_2\right) \subset \widehat{{\mathfrak{M}}}_1\times \widehat{{\mathfrak{M}}}_2. {\tag{11.2.2}} \end{equation}$$

Let

$$\begin{equation*} \widehat{{\mathfrak{P}}}_{k}^{(n)} \overset{\operatorname {\scriptstyle def.}}{=}\widetilde{{\mathfrak{P}}}_{k}^{(n)\circ }\cap \left(\widehat{{\mathfrak{M}}}_1\times \widehat{{\mathfrak{M}}}_2\right). \end{equation*}$$

The quotient $V^2_k/V^1_k$ defines vector bundles over $\widetilde{{\mathfrak{P}}}_{k}^{(n)}$, $\widetilde{{\mathfrak{P}}}_{k}^{(n)\circ }$ and $\widehat{{\mathfrak{P}}}_{k}^{(n)}$. For brevity, all are simply denoted by $V^2_k/V^1_k$.

Let us denote by $i_a$ the inclusion $\widehat{{\mathfrak{M}}}_a\subset \widetilde{{\mathfrak{M}}}^\circ _a$ for $a=1,2$. By 11.2.2, the inclusion map $i_1\times \operatorname {id}_{\widetilde{{\mathfrak{M}}}^\circ ({\mathbf{v}}^2,{\mathbf{w}})}\colon \widehat{{\mathfrak{M}}}({\mathbf{v}}^1,{\mathbf{w}})\times \widetilde{{\mathfrak{M}}}^\circ ({\mathbf{v}}^2,{\mathbf{w}})\to \widetilde{{\mathfrak{M}}}^\circ ({\mathbf{v}}^1,{\mathbf{w}})\times \widetilde{{\mathfrak{M}}}^\circ ({\mathbf{v}}^2,{\mathbf{w}})$ induces the pull-back homomorphism with support

$$\begin{equation*} (i_1\times \operatorname {id}_{\widetilde{{\mathfrak{M}}}^\circ _2})^* \colon K^{G'\times G_{\mathbf{w}}\times {\mathbb{C}}^*} (\widetilde{{\mathfrak{P}}}_{k}^{(n)\circ })\to K^{G'\times G_{\mathbf{w}}\times {\mathbb{C}}^*}(\widehat{{\mathfrak{P}}}_{k}^{(n)}). \end{equation*}$$

Similarly, we have

$$\begin{equation*} (\operatorname {id}_{\widetilde{{\mathfrak{M}}}^\circ _1}\times i_2)^* \colon K^{G'\times G_{\mathbf{w}}\times {\mathbb{C}}^*} (\widetilde{{\mathfrak{P}}}_{k}^{(n)\circ })\to K^{G'\times G_{\mathbf{w}}\times {\mathbb{C}}^*}(\widehat{{\mathfrak{P}}}_{k}^{(n)}). \end{equation*}$$

Lemma 11.2.3.

We have

$$\begin{equation*} (i_1\times \operatorname {id}_{\widetilde{{\mathfrak{M}}}^\circ _2})^* [\mathcal{O}_{\widetilde{{\mathfrak{P}}}_{k}^{(n)\circ }}] = [\mathcal{O}_{\widehat{{\mathfrak{P}}}_{k}^{(n)}}], \qquad (\operatorname {id}_{\widetilde{{\mathfrak{M}}}^\circ _2}\times i_2)^* [\mathcal{O}_{\widetilde{{\mathfrak{P}}}_{k}^{(n)\circ }}] = [\mathcal{O}_{\widehat{{\mathfrak{P}}}_{k}^{(n)}}]. \end{equation*}$$

More generally, if $T(V^2_k/V^1_k)$ denotes a tensor product of exterior products of the bundle $V^2_k/V^1_k$ and its dual, we have

$$\begin{equation*} \begin{split} & (i_1\times \operatorname {id}_{\widetilde{{\mathfrak{M}}}^\circ _2})^* [T \left(V^2_k/V^1_k\right)] = [T \left(V^2_k/V^1_k\right)\otimes \mathcal{O}_{\widehat{{\mathfrak{P}}}_{k}^{(n)}}],\\ & (\operatorname {id}_{\widetilde{{\mathfrak{M}}}^\circ _1}\times i_2)^* [T \left(V^2_k/V^1_k\right)] = [T \left(V^2_k/V^1_k\right)\otimes \mathcal{O}_{\widehat{{\mathfrak{P}}}_{k}^{(n)}}]. \end{split} \end{equation*}$$

Proof.

The latter statement follows from the first statement and the formula 6.4.1. Thus it is enough to check the first statement, and the first statement follows from the transversality of intersections 11.2.2 in $\widetilde{{\mathfrak{M}}}^\circ _1\times \widetilde{{\mathfrak{M}}}^\circ _2$.

Let $\mu '\colon {\mathbf{M}}({\mathbf{v}},{\mathbf{w}})\to \bigoplus _{l\neq k}{\mathfrak{gl}}(V_l)$ be the $\bigoplus _{l\neq k}{\mathfrak{gl}}(V_l)$-part of the moment map $\mu$. It induces a map $\widetilde{{\mathfrak{M}}}^\circ _a\to \bigoplus _{l\neq k}{\mathfrak{gl}}(V_l)$ for $a=1, 2$. Let us denote it by $\mu _a'$. Thus we have $\widehat{{\mathfrak{M}}}_a = \mu _a^{\prime -1}(0)$. Composing $\mu _a'$ with the projection $p_a\colon \widetilde{{\mathfrak{M}}}^\circ _1\times \widetilde{{\mathfrak{M}}}^\circ _2\to \widetilde{{\mathfrak{M}}}^\circ _a$, we have a map $\mu _a'\circ p_a\colon \widetilde{{\mathfrak{M}}}^\circ _1\times \widetilde{{\mathfrak{M}}}^\circ _2\to \bigoplus _{l\neq k}{\mathfrak{gl}}(V_l)$ for $a=1,2$. We denote it also by $\mu _a'$ for brevity. It is enough to show that the restriction of the differential $d\mu _a'$ to $T\widetilde{{\mathfrak{P}}}_k^{(n)\circ }$ is surjective on $\widehat{{\mathfrak{P}}}_k^{(n)} = \widetilde{{\mathfrak{P}}}_k^{(n)\circ }\cap (\widetilde{{\mathfrak{M}}}^\circ _1\times \widehat{{\mathfrak{M}}}_2) = \widetilde{{\mathfrak{P}}}_k^{(n)\circ }\cap (\widehat{{\mathfrak{M}}}_1\times \widetilde{{\mathfrak{M}}}^\circ _2)$.

We consider the homomorphisms $\sigma _k$, $\tau _k$ defined in 2.9.1 where $(B,i,j)$ is replaced by $(B^a,i^a,j^a)$ ($a=1,2$). We denote them by $\sigma ^a_k$ and $\tau ^a_k$ respectively.

Take a point $([B^1,i^1,j^1],[B^2,i^2,j^2])\in \widehat{{\mathfrak{P}}}_k^{(n)}$. Then

$$\begin{equation*} B^1_h = B^2_h \quad (\operatorname {in}(h)\neq k, \operatorname {out}(h)\neq k), \quad i^1_l = i^2_l, \quad j^1_l = j^2_l \quad (l\neq k), \end{equation*}$$

and there exists $\xi _k\colon V_k^1\to V_k^2$ such that

$$\begin{equation*} \xi _k B^1_h = B^2_h, \quad B^1_{\overline{h}} \xi _k = B^2_{\overline{h}} \quad (\operatorname {in}(h) = k),\quad \xi _k i^1_k = i^2_k, \quad j^1_k = j^2_k \xi _k. \end{equation*}$$

The tangent space $T\widetilde{{\mathfrak{P}}}_k^{(n)\circ }$ is isomorphic to the space of $(\delta B^1, \delta i^1, \delta j^1, \delta B^2, \delta i^2, \delta j^2) \in {\mathbf{M}}({\mathbf{v}}^1,{\mathbf{w}})\times {\mathbf{M}}({\mathbf{v}}^2,{\mathbf{w}})$ such that

$$\begin{equation} \begin{gathered} \delta B^1_h = \delta B^2_h \quad (\text{if $\operatorname {in}(h)\neq k$, $\operatorname {out}(h)\neq k$}), \\ \delta i_l^1 = \delta i_l^2,\quad \delta j_l^1 = \delta j_l^2 \quad (\text{for $l\neq k$}),\\ \tau _k^a\, \delta \sigma _k^a + \delta \tau _k^a\, \sigma _k^a = 0 \qquad (a=1,2),\\ \xi _k\circ \delta \tau _k^1 = \delta \tau _k^2 , \qquad \delta \sigma _k^1 = \delta \sigma _k^2\circ \xi _k \end{gathered} {\tag{11.2.4}} \end{equation}$$

modulo the image of

$$\begin{equation} \begin{split} & \{ (\delta \xi _k^1, \delta \xi _k^2) \in {\mathfrak{gl}}(V_k^1)\times {\mathfrak{gl}}(V^2_k)\mid \delta \xi _k^2\, \xi _k = \xi _k \delta \xi _k^1 \} \\ & \qquad \longmapsto \left\{ \begin{aligned} &\delta \xi _k^1 B^1_h, \; - B^1_{\overline{h}}\delta \xi _k^1, \; \delta \xi _k^2 B^2_h, \; - B^2_{\overline{h}}\delta \xi _k^2 \quad (\operatorname {in}(h) = k), \\ & \delta \xi _k^1 i_k^1, \; -j_k^1 \delta \xi _k^1,\; \delta \xi _k^2 i_k^2, \; -j_k^2 \delta \xi _k^2\\ & \text{other components are $0$}, \end{aligned} \right. \end{split} {\tag{11.2.5}} \end{equation}$$

where

$$\begin{equation*} \delta \sigma _k^a = \bigoplus _{\operatorname {in}(h)=k}\delta B_{\overline{h}}^a\oplus \delta j_k^a, \quad \delta \tau _k^a = \bigoplus _{\operatorname {in}(h)=k}\varepsilon (h)\delta B_h^a + \delta i_k^a \quad (a=1,2), \end{equation*}$$

and we have used the identification $V^1_l \cong V^2_l$ for $l\neq k$.

Now suppose that $(\zeta _l)_{l\neq k} \in \bigoplus _{l\neq k} {\mathfrak{gl}}(V_l)$ is orthogonal to the image of $d\mu '_a|_{T\widetilde{{\mathfrak{P}}}_k^{(n)\circ }}$. Putting $\zeta _k = 0$, we consider $\zeta = (\zeta _l)$ as an element of $\operatorname {L}(V^a,V^a)$. Then

$$\begin{equation*} \operatorname {tr}\left(\varepsilon \delta B^a(B^a\zeta - \zeta B^a) + \delta i^a\, j^a\zeta + \zeta i^a\, \delta j^a\right) = 0 \end{equation*}$$

for any $(\delta B^1, \delta i^1, \delta j^1, \delta B^2, \delta i^2, \delta j^2)\in T\widetilde{{\mathfrak{P}}}_k^{(n)\circ }$. Since the image of 11.2.5 lies in the kernel of $d\mu '_a$, the above equality holds for any $(\delta B^1, \delta i^1, \delta j^1, \delta B^2, \delta i^2, \delta j^2)$ satisfying 11.2.4.

Taking $(\delta B^1, \delta i^1, \delta j^1, \delta B^2, \delta i^2, \delta j^2)$ from $\Delta {\mathbf{M}}'({\mathbf{v}},{\mathbf{w}})$, we find

$$\begin{equation} \begin{gathered} B_h^a\zeta _{\operatorname {out}(h)} = \zeta _{\operatorname {in}(h)} B_h^a\qquad \text{if $\operatorname {in}(h)\neq k$, $\operatorname {out}(h)\neq k$},\\ \zeta _l i^2_l = 0, \quad j_l^2 \zeta _l = 0\qquad \text{if $l\neq k$}. \end{gathered} {\tag{11.2.6}} \end{equation}$$

Next taking $(\delta B^1, \delta i^1, \delta j^1, \delta B^2, \delta i^2, \delta j^2)$ from the other component (data related to the vertex $k$), we get

$$\begin{align*} & \sigma _k^a \circ \left(\sum _{\operatorname {in}(h)=k} \varepsilon (h) B^a_h \zeta _{\operatorname {out}(h)}, 0\right)\\ &\qquad = \left( \bigoplus _{\operatorname {in}(h)=k} \zeta _{\operatorname {out}(h)} B^a_{\overline{h}} \oplus 0 \right)\circ \tau _k^a \in \operatorname {End}\left(\bigoplus _{\operatorname {in}(h)=k} V_{\operatorname {out}(h)}\oplus W_k\right). \end{align*}$$

Comparing $\operatorname {Hom}(V_{\operatorname {out}(h)}, W_k)$-components, we find

$$\begin{equation} \varepsilon (h) j_k B^a_h \zeta _{\operatorname {out}(h)} = 0. {\tag{11.2.7}} \end{equation}$$

Comparing $\operatorname {Hom}(V_{\operatorname {out}(h)}, V_{\operatorname {out}(h')})$-components, we have

$$\begin{equation} B^a_{\overline{h'}}\varepsilon (h) B^a_h \zeta _{\operatorname {out}(h)} = \zeta _{\operatorname {out}(h')} B^a_{\overline{h'}}\varepsilon (h) B^a_h. {\tag{11.2.8}} \end{equation}$$

If we define

$$\begin{equation*} S_l \overset{\operatorname {\scriptstyle def.}}{=}\begin{cases} \operatorname {Im}\zeta _l & \text{if $l\neq k$}, \\ \sum _{\operatorname {in}(h)=k} \operatorname {Im}\left(B^a_h \zeta _{\operatorname {out}(h)}\right) &\text{if $l = k$}, \end{cases} \end{equation*}$$

11.2.6, 11.2.7 and 11.2.8 imply that $S = (S_l)$ is $B^a$-invariant and contained in $\operatorname {Ker}j$. Thus $S = 0$ by the stability condition. In particular, we have $\zeta = 0$. This means that $d\mu '_a|_{T\widetilde{{\mathfrak{P}}}_k^{(n)\circ }}$ is surjective.

■

Let $\pi _a\colon \widehat{{\mathfrak{M}}}_a\to {\mathfrak{M}}_a$ be the projection ($a = 1,2$). Then we have

$$\begin{equation} \begin{gathered} \text{the restriction of $\operatorname {id}_{\widehat{{\mathfrak{M}}}_1}\times \,\pi _2\colon \widehat{{\mathfrak{M}}}_1\times \widehat{{\mathfrak{M}}}_2 \to \widehat{{\mathfrak{M}}}_1\times {\mathfrak{M}}_2$ to $\widehat{{\mathfrak{P}}}_k^{(n)}$ is proper,} \\ (\operatorname {id}_{\widehat{{\mathfrak{M}}}_1}\times \,\pi _2)(\widehat{{\mathfrak{P}}}_k^{(n)}) = (\pi _1\times \operatorname {id}_{{\mathfrak{M}}_2})^{-1}({\mathfrak{P}}_k^{(n)}), \\ \text{the restriction of $\pi _1\times \operatorname {id}_{\widehat{{\mathfrak{M}}}_2}\colon \widehat{{\mathfrak{M}}}_1\times \widehat{{\mathfrak{M}}}_2 \to {\mathfrak{M}}_1\times \widehat{{\mathfrak{M}}}_2$ to $\widehat{{\mathfrak{P}}}_k^{(n)}$ is proper,} \\ (\pi _1\times \operatorname {id}_{\widehat{{\mathfrak{M}}}_2})(\widehat{{\mathfrak{P}}}_k^{(n)}) = (\operatorname {id}_{{\mathfrak{M}}_1}\times \,\pi _2)^{-1}({\mathfrak{P}}_k^{(n)}). \end{gathered} {\tag{11.2.9}} \end{equation}$$

By these properties, we have homomorphisms

$$\begin{align*} & (\pi _1\times \operatorname {id}_{{\mathfrak{M}}_2})^{*-1} \left(\operatorname {id}_{\widehat{{\mathfrak{M}}}_1}\times \,\pi _2\right)_*\colon K^{G'\times G_{\mathbf{w}}\times {\mathbb{C}}^*}(\widehat{{\mathfrak{P}}}_k^{(n)})\to K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}({\mathfrak{P}}_k^{(n)}), \\ & (\operatorname {id}_{{\mathfrak{M}}_1}\times \,\pi _1)^{*-1} \left(\pi _1\times \operatorname {id}_{\widehat{{\mathfrak{M}}}_2}\right)_*\colon K^{G'\times G_{\mathbf{w}}\times {\mathbb{C}}^*}(\widehat{{\mathfrak{P}}}_k^{(n)})\to K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}({\mathfrak{P}}_k^{(n)}). \end{align*}$$

Lemma 11.2.10.

We have

$$\begin{equation*} \begin{split} & (\pi _1\times \operatorname {id}_{{\mathfrak{M}}_2})^{*-1} \left(\operatorname {id}_{\widehat{{\mathfrak{M}}}_1}\times \,\pi _2\right)_* [\mathcal{O}_{\widehat{{\mathfrak{P}}}_k^{(n)}}] = [\mathcal{O}_{{\mathfrak{P}}_k^{(n)}}], \\ & (\operatorname {id}_{{\mathfrak{M}}_1}\times \,\pi _2)^{*-1} \left(\pi _1\times \operatorname {id}_{\widehat{{\mathfrak{M}}}_2}\right)_* [\mathcal{O}_{\widehat{{\mathfrak{P}}}_k^{(n)}}] = [\mathcal{O}_{{\mathfrak{P}}_k^{(n)}}]. \end{split} \end{equation*}$$

More generally, if $T(V^2_k/V^1_k)$ denotes a tensor product of exterior products of the bundle $V^2_k/V^1_k$ and its dual, we have

$$\begin{align*} & (\pi _1\times \operatorname {id}_{{\mathfrak{M}}_2})^{*-1} \left(\operatorname {id}_{\widehat{{\mathfrak{M}}}_1}\times \,\pi _2\right)_* [(\pi _1\times \,\pi _2)^* T\left( V^2_k/V^1_k\right)] = [T\left( V^2_k/V^1_k\right)], \\ & (\operatorname {id}_{{\mathfrak{M}}_1}\times \,\pi _2)^{*-1} \left(\pi _1\times \operatorname {id}_{\widehat{{\mathfrak{M}}}_2}\right)_* [(\pi _1\times \,\pi _2)^* T\left(V^2_k/V^1_k\right)] = [T\left(V^2_k/V^1_k\right)]. \end{align*}$$

Proof.

The latter statement follows from the former one together with the projection formula 6.5.1. Thus it is enough to prove the former statement.

By definition, $(\pi _1\times \operatorname {id}_{{\mathfrak{M}}_2})^{-1}({\mathfrak{P}}_k^{(n)})$ consists of

$$\begin{equation*} (\operatorname {GL}(V^1_k)\cdot (B^1,i^1,j^1), G_{{\mathbf{v}}^2}\cdot (B^2,i^2,j^2)) \in \widehat{{\mathfrak{M}}}_1\times {\mathfrak{M}}_2 \end{equation*}$$

such that there exists $\xi \in \operatorname {L}(V^1, V^2)$ satisfying 5.1.3. We fix representatives $(B^1,i^1,j^1)$, $(B^2,i^2,j^2)$. Then the above $\xi$ is uniquely determined. Recall that we have chosen the identification $V^1_l\cong V^2_l$ for $l\neq k$ over $\widehat{{\mathfrak{M}}}_1\times \widehat{{\mathfrak{M}}}_2$. Let us define $\xi '\in \operatorname {L}(V^2,V^2)$ by

$$\begin{equation*} \xi '_l \overset{\operatorname {\scriptstyle def.}}{=}\begin{cases} \operatorname {id}& \text{if $l = k$}, \\ \xi _l & \text{otherwise}. \end{cases} \end{equation*}$$

We define a new datum

$$\begin{equation*} (B^3, i^3, j^3) \overset{\operatorname {\scriptstyle def.}}{=}(\xi ^{\prime -1}B^2 \xi ', \xi ^{\prime -1}i^2, j^2\xi '). \end{equation*}$$

By definition, we have

$$\begin{gather*} \xi _k B^1_h = B^3_h, \quad B^1_{\overline{h}} = B^3_{\overline{h}}\xi _k \quad (\operatorname {in}(h) = k), \quad \xi _k i^1_k = i^3_k , \quad j^1_k = j^3_k\xi _k, \\ B^1_h = B^3_h \quad (\operatorname {in}(h)\neq k, \operatorname {out}(h)\neq k), \quad i^1_l = i^3_l, \quad j^1_l = j^3_l \quad (l\neq k). \end{gather*}$$

Hence $(\operatorname {GL}(V^1_k)\cdot (B^1,i^1,j^1), \operatorname {GL}(V^2_k)\cdot (B^3,i^3,j^3))$ is contained in $\widehat{{\mathfrak{P}}}_k$. Moreover, $\operatorname {GL}(V^2_k)\cdot (B^3,i^3,j^3)$ is independent of the choice of the representative $(B^2,i^2,j^2)$. Thus we have defined a map $(\pi _1\times \operatorname {id}_{{\mathfrak{M}}_2})^{-1}({\mathfrak{P}}_k^{(n)})\to \widehat{{\mathfrak{P}}}_k^{(n)}$ by

$$\begin{align*} &(\operatorname {GL}(V^1_k)\cdot (B^1,i^1,j^1), G_{{\mathbf{v}}^2}\cdot (B^2,i^2,j^2))\\ &\qquad \mapsto (\operatorname {GL}(V^1_k)\cdot (B^1,i^1,j^1), \operatorname {GL}(V^2_k)\cdot (B^3,i^3,j^3)), \end{align*}$$

which is the inverse of the restriction of $\operatorname {id}_{\widetilde{{\mathfrak{M}}}_1}\times \pi _2$. In particular, this implies

$$\begin{equation*} \left(\operatorname {id}_{\widehat{{\mathfrak{M}}}_1}\times \,\pi _2\right)_* [\mathcal{O}_{\widehat{{\mathfrak{P}}}_k^{(n)}}] = [\mathcal{O}_{(\pi _1\times \operatorname {id}_{{\mathfrak{M}}_2})^{-1}({\mathfrak{P}}_k^{(n)})}]. \end{equation*}$$

Since $\pi _1\times \operatorname {id}_{{\mathfrak{M}}_2}\colon (\pi _1\times \operatorname {id}_{{\mathfrak{M}}_2})^{-1}({\mathfrak{P}}_k^{(n)})\to {\mathfrak{P}}_k^{(n)}$ is a principal $G'$-bundle, we have

$$\begin{equation*} (\pi _1\times \operatorname {id}_{{\mathfrak{M}}_2})^*[\mathcal{O}_{{\mathfrak{P}}_k^{(n)}}] = [\mathcal{O}_{(\pi _1\times \operatorname {id}_{{\mathfrak{M}}_2})^{-1}({\mathfrak{P}}_k^{(n)})}]. \end{equation*}$$

Thus we have proved the first equation. The second equation can be proved in a similar way.

■

11.3. Reduction to the rank $1$ case

First consider the relation 1.2.8 for $k = l$. Let ${\mathbf{v}}^1$, ${\mathbf{v}}^2$, ${\mathbf{v}}^3$, ${\mathbf{v}}^4$ be dimension vectors such that

$$\begin{equation*} {\mathbf{v}}^1 = {\mathbf{v}}^3, \quad {\mathbf{v}}^2 = {\mathbf{v}}^1 + \alpha _k, \quad {\mathbf{v}}^4 = {\mathbf{v}}^1 - \alpha _k. \end{equation*}$$

We want to compute $x_{k}^+(z) * x_{k}^-(w)$ and $x_{k}^-(w) * x_{k}^+(z)$ in the component

$$\begin{equation*} K^{G_{\mathbf{w}}\times {\mathbb{C}}}(Z({\mathbf{v}}^1,{\mathbf{v}}^3,{\mathbf{w}})), \end{equation*}$$

and then compare it with the right hand side of 1.2.8 with $k = l$ in the same component.

Let $\widehat{{\mathfrak{M}}}({\mathbf{v}}^a,{\mathbf{w}})$, $\widetilde{{\mathfrak{M}}}({\mathbf{v}}^a,{\mathbf{w}})$, $\widetilde{{\mathfrak{M}}}^\circ ({\mathbf{v}}^a,{\mathbf{w}})$, $G'_{{\mathbf{v}}^a}$ and ${\mathbf{M}}'({\mathbf{v}}^a,{\mathbf{w}})$ be as in §11.1. Let ${\mathfrak{P}}_k({\mathbf{v}}^a,{\mathbf{w}})$, $\widehat{{\mathfrak{P}}}_k({\mathbf{v}}^a,{\mathbf{w}})$, $\widetilde{{\mathfrak{P}}}_k({\mathbf{v}}^a,{\mathbf{w}})$, $\widetilde{{\mathfrak{P}}}_k^\circ ({\mathbf{v}}^a,{\mathbf{w}})$ be the Hecke correspondence and its modifications introduced in §11.2. (We drop the superscript $(n)$ and write the dimension vector ${\mathbf{v}}^a$, ${\mathbf{w}}$.) Let $\omega$ be the exchange of factors as before. Let $\widehat{Z}({\mathbf{v}}^a,{\mathbf{v}}^b;{\mathbf{w}})$, $\widetilde{Z}({\mathbf{v}}^a,{\mathbf{v}}^b;{\mathbf{w}})$, $\widetilde{Z}^\circ ({\mathbf{v}}^a,{\mathbf{v}}^b;{\mathbf{w}})$ be subvarieties in $\widehat{{\mathfrak{M}}}({\mathbf{v}}^a,{\mathbf{w}})\times \widehat{{\mathfrak{M}}}({\mathbf{v}}^b,{\mathbf{w}})$, $\widetilde{{\mathfrak{M}}}({\mathbf{v}}^a,{\mathbf{w}})\times \widetilde{{\mathfrak{M}}}({\mathbf{v}}^b,{\mathbf{w}})$, $\widetilde{{\mathfrak{M}}}^\circ ({\mathbf{v}}^a,{\mathbf{w}})\times \widetilde{{\mathfrak{M}}}^\circ ({\mathbf{v}}^b,{\mathbf{w}})$ defined in the same way as $Z({\mathbf{v}}^a,{\mathbf{v}}^b;{\mathbf{w}})$.

We have the following commutative diagram:

$$\begin{equation*} \begin{CD} {K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}({\mathfrak{P}}_k({\mathbf{v}}^2,{\mathbf{w}}))\times K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(\omega {\mathfrak{P}}_k({\mathbf{v}}^2,{\mathbf{w}}))} @>>> {K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(Z({\mathbf{v}}^1,{\mathbf{v}}^3;{\mathbf{w}}))} \\@AAA @AAA \\{K^{G'\times G_{\mathbf{w}}\times {\mathbb{C}}^*}(\widehat{{\mathfrak{P}}}_k({\mathbf{v}}^2,{\mathbf{w}}))\times K^{G'\times G_{\mathbf{w}}\times {\mathbb{C}}^*}(\omega \widehat{{\mathfrak{P}}}_k({\mathbf{v}}^2,{\mathbf{w}}))} @>>> {K^{G'\times G_{\mathbf{w}}\times {\mathbb{C}}^*}(\widehat{Z}({\mathbf{v}}^1,{\mathbf{v}}^3;{\mathbf{w}}))} \\@AAA @AAA \\{K^{G'\times G_{\mathbf{w}}\times {\mathbb{C}}^*}(\widetilde{{\mathfrak{P}}}_k^\circ ({\mathbf{v}}^2,{\mathbf{w}}))\times K^{G'\times G_{\mathbf{w}}\times {\mathbb{C}}^*}(\omega \widetilde{{\mathfrak{P}}}_k^\circ ({\mathbf{v}}^2,{\mathbf{w}}))} @>>> {K^{G'\times G_{\mathbf{w}}\times {\mathbb{C}}^*}(\widetilde{Z}^\circ ({\mathbf{v}}^1,{\mathbf{v}}^3;{\mathbf{w}}))} \\@AAA @AAA \\{K^{G'\times G_{\mathbf{w}}\times {\mathbb{C}}^*}(\widetilde{{\mathfrak{P}}}_k({\mathbf{v}}^2,{\mathbf{w}}))\times K^{G'\times G_{\mathbf{w}}\times {\mathbb{C}}^*}(\omega \widetilde{{\mathfrak{P}}}_k({\mathbf{v}}^2,{\mathbf{w}}))} @>>> {K^{G'\times G_{\mathbf{w}}\times {\mathbb{C}}^*} (\widetilde{Z}({\mathbf{v}}^1,{\mathbf{v}}^3;{\mathbf{w}}))}. \end{CD} \end{equation*}$$

The horizontal arrows are convolution products relative to

(1)

${\mathfrak{M}}({\mathbf{v}}^1,{\mathbf{w}})$, ${\mathfrak{M}}({\mathbf{v}}^2,{\mathbf{w}})$, ${\mathfrak{M}}({\mathbf{v}}^3,{\mathbf{w}})$,

(2)

$\widehat{{\mathfrak{M}}}({\mathbf{v}}^1,{\mathbf{w}})$, $\widehat{{\mathfrak{M}}}({\mathbf{v}}^2,{\mathbf{w}})$, $\widehat{{\mathfrak{M}}}({\mathbf{v}}^3,{\mathbf{w}})$,

(3)

$\widetilde{{\mathfrak{M}}}^\circ ({\mathbf{v}}^1,{\mathbf{w}})$, $\widetilde{{\mathfrak{M}}}^\circ ({\mathbf{v}}^2,{\mathbf{w}})$, $\widetilde{{\mathfrak{M}}}^\circ ({\mathbf{v}}^3,{\mathbf{w}})$,

(4)

$\widetilde{{\mathfrak{M}}}({\mathbf{v}}^1,{\mathbf{w}})$, $\widetilde{{\mathfrak{M}}}({\mathbf{v}}^2,{\mathbf{w}})$, $\widetilde{{\mathfrak{M}}}({\mathbf{v}}^3,{\mathbf{w}})$.

The vertical arrows between the first and the second rows are homomorphisms given in Proposition 8.3.5. The arrows between the second and the third are homomorphisms given in Proposition 8.2.3. By the properties 11.2.2, 11.2.9 and

(a)

$\widetilde{Z}^\circ ({\mathbf{v}}^1,{\mathbf{v}}^3;{\mathbf{w}})\cap \left(\widehat{{\mathfrak{M}}}({\mathbf{v}}^1,{\mathbf{w}})\times \widetilde{{\mathfrak{M}}}^\circ ({\mathbf{v}}^2,{\mathbf{w}})\right) \subset \widehat{Z}({\mathbf{v}}^1,{\mathbf{v}}^3;{\mathbf{w}})$,

(b)

the restriction of $\pi _1\times \operatorname {id}_{\widehat{{\mathfrak{M}}}({\mathbf{v}}^3,{\mathbf{w}})}\colon \widehat{{\mathfrak{M}}}({\mathbf{v}}^1,{\mathbf{w}})\times \widehat{{\mathfrak{M}}}({\mathbf{v}}^3,{\mathbf{w}})\to {\mathfrak{M}}({\mathbf{v}}^1,{\mathbf{w}})\times \widehat{{\mathfrak{M}}}({\mathbf{v}}^3,{\mathbf{w}})$ to $\widehat{Z}({\mathbf{v}}^1,{\mathbf{v}}^3;{\mathbf{w}})$ is proper,

(c)

$(\pi _1\times \operatorname {id}_{\widehat{{\mathfrak{M}}}({\mathbf{v}}^3,{\mathbf{w}})})(\widehat{Z}({\mathbf{v}}^1,{\mathbf{v}}^3;{\mathbf{w}})) \subset (\operatorname {id}_{{\mathfrak{M}}({\mathbf{v}}^1,{\mathbf{w}})}\times \,\pi _3)^{-1}(Z({\mathbf{v}}^1,{\mathbf{v}}^3;{\mathbf{w}}))$,

those homomorphisms can be defined. Finally the arrows between the third and the fourth are restrictions to open subvarieties.

The commutativity for the first and the second squares follows from Propositions 8.3.5 and 8.2.3 respectively. The last square is also commutative since $\widetilde{{\mathfrak{M}}}^\circ ({\mathbf{v}}^a,{\mathbf{w}})$ is an open subvariety of $\widetilde{{\mathfrak{M}}}({\mathbf{v}}^a,{\mathbf{w}})$ and since we have 11.2.1.

Recall that the modified Hecke correspondence in the last row is the product of the Hecke correspondence for type $A_1$ and the diagonal $\Delta {\mathbf{M}}'({\mathbf{v}}^1,{\mathbf{w}})$. Under the composite of vertical homomorphisms, $e_{k,r}$, $f_{k,s}$ at the upper left are the images of the exterior products of the corresponding elements for type $A_1$ and $\mathcal{O}_{\Delta {\mathbf{M}}'({\mathbf{v}}^1,{\mathbf{w}})}$ at the lower left, except for the following two differences:

(a)

the groups acting on varieties are different,

(b)

the sign factors in 9.3.2, which involve the orientation $\Omega$, are different.

For the quiver varieties of type $A_1$, the group is

$$\begin{equation*} \operatorname {GL}\left(\bigoplus _{h:\operatorname {out}(h)=k} V_{\operatorname {in}(h)}\oplus W_k\right) \times {\mathbb{C}}^* \cong \operatorname {GL}_N({\mathbb{C}})\times {\mathbb{C}}^*. \end{equation*}$$

But, if we define a homomorphism $G'\times G_{\mathbf{w}}\times {\mathbb{C}}^* \to \operatorname {GL}_N({\mathbb{C}})\times {\mathbb{C}}^*$ by

$$\begin{equation*} \left((g_l)_{l\in I:l\neq k}, (h_{l'})_{l'\in I}, q\right) \longmapsto \left(\bigoplus _{h:\operatorname {out}(h)=k} q^{m(h)} g_{\operatorname {in}(h)} \oplus h_k, q\right), \end{equation*}$$

we have an induced homomorphism in equivariant $K$-groups: $K^{\operatorname {GL}_N({\mathbb{C}})\times {\mathbb{C}}^*}(\ )\to K^{G'\times G_{\mathbf{w}}\times {\mathbb{C}}^*}(\ )$. (Here $m(h)$ is as in 2.7.1.) It is compatible with the convolution product, hence it is enough to check the relation in $K^{\operatorname {GL}_N({\mathbb{C}})\times {\mathbb{C}}^*}(\ )$.

Furthermore, the sign factor cancels out in $e_{k,r} * f_{k,s}$. Thus the above differences make no effect when we check the relation 1.2.8.

By the commutativity of the diagram, $e_{k,r} * f_{k,s}$ is the image of the corresponding element in the lower right.

We have a similar commutative diagram to compute $f_{k,s} * e_{k,r}$. Hence the commutator $[e_{k,r}, f_{k,s}]$ is the image of the corresponding commutator in the lower right. In the next section, we will check the relation 1.2.8 for type $A_1$. In particular, the commutator in the lower right is represented by tautological bundles, considered as an element of the $K$-theory of the diagonal $\Delta \widetilde{{\mathfrak{M}}}({\mathbf{v}}^1,{\mathbf{w}})$. Note that $\mathcal{O}_{\Delta \widetilde{{\mathfrak{M}}}({\mathbf{v}}^1,{\mathbf{w}})}$ is mapped to $\mathcal{O}_{\Delta {\mathfrak{M}}({\mathbf{v}}^1,{\mathbf{w}})}$ by examples in §8, and that the tautological bundles on $\widetilde{{\mathfrak{M}}}({\mathbf{v}}^1,{\mathbf{w}})$ are restricted to tautological bundles on ${\mathfrak{M}}({\mathbf{v}}^1,{\mathbf{w}})$. Hence we have exactly the same relation 1.2.8 for the general case.

Similarly, we can reduce the check of the relation 1.2.9 to the case of type $A_1$.

11.4. Rank $1$ case

In this subsection, we check the relation when the graph is of type $A_1$. This calculation is essentially the same as the one by Vasserot 58, but we reproduce it here for the convenience of the reader. (Remark that our ${\mathbb{C}}^*$-action is different from the one in 58. The definition of $e_{k,r}$, etc. is also different.) We drop the subscript $k\in I$ as usual.

We prepare several notations. For $a,b\in {\mathbb{Z}}$, let

$$\begin{equation*} [a,b] \overset{\operatorname {\scriptstyle def.}}{=}\begin{cases} \{ a, a+1, \dots , b \} & \text{if $b \ge a$}, \\ \emptyset & \text{otherwise}. \end{cases} \end{equation*}$$

Let

$$\begin{equation*} {\mathbf{R}} \overset{\operatorname {\scriptstyle def.}}{=}{\mathbb{Z}}[q,q^{-1}][x_1^{\pm },\dots , x_{N}^\pm ]. \end{equation*}$$

For a partition $I = (I_1, I_2)$ of the set $\{1, \dots , N\}$ into $2$ subsets, let $S_I = S_{I_1} \times S_{I_2}$ be the subgroups of $S_N$ consisting of permutations which preserve each subset. For a subgroup $G\subset S_N$, let ${\mathbf{R}}^{G}$ be the subring of ${\mathbf{R}}$ consisting of elements which are fixed by the action of $G$. If $J$ is another partition of $\{1,\dots ,N\}$, we define the symmetrizer ${\mathfrak{S}}_I^J\colon {\mathcal{R}}^{S_I\cap S_J}\to {\mathcal{R}}^{S_J}$ by

$$\begin{equation*} f \mapsto \sum _{\sigma \in S_J/S_I\cap S_J} \sigma (f), \end{equation*}$$

where $\mathcal{R}$ is the quotient field of ${\mathbf{R}}$. For each $v\in [0, N]$, let $[v]$ be the partition $([1, v], [v+1,N])$. If $I = (I_1, I_2)$ is a partition of $\{1,\dots , N\}$ into $2$ subsets and $k\in I_1$ (resp. $k\in I_2$), we define a new partition $\tau _k^+(I)$ (resp. $\tau _k^-(I)$) by

$$\begin{equation*} \tau _k^+(I) \overset{\operatorname {\scriptstyle def.}}{=}(I_1\setminus \{k\}, I_2\cup \{ k\}) \quad \left(\text{resp. $\tau _k^-(I)\overset{\operatorname {\scriptstyle def.}}{=}(I_1\cup \{ k\}, I_2\setminus \{k\})$}\right). \end{equation*}$$

If $I = (I_1, I_2)$ is a partition and $f\in {\mathbf{R}}^{S_{[v]}}$, we define

$$\begin{equation*} f(x_I) \overset{\operatorname {\scriptstyle def.}}{=}f(x_{i_1}, \dots , x_{i_v}, x_{j_1}, \dots , x_{j_{N-v}}), \end{equation*}$$

where $I_1 = \{ i_1, \dots , i_v \}$, $I_2 = \{ j_1, \dots , j_{N-v} \}$.

Let ${\mathfrak{M}}(v,N)$ be the quiver variety for the graph of type $A_1$ with dimension vectors $v$, $N$. It is isomorphic to the cotangent bundle of the Grassmann variety of $v$-dimensional subspaces of an $N$-dimensional space. Let $G(v,N)$ denote the Grassmann variety contained in ${\mathfrak{M}}(v,N)$ as the $0$-section. Let $Z(v^1,v^2;N)$ be the analogue of Steinberg’s variety as before. The following lemma is crucial.

Lemma 11.4.1 (58, Lemma 13, 13, Claim 7.6.7).

The representation of

$$\begin{equation*} \bigoplus K^{\operatorname {GL}_N({\mathbb{C}})\times {\mathbb{C}}^*}(Z(v^1,v^2;N)) \end{equation*}$$

on $\bigoplus K^{\operatorname {GL}_N({\mathbb{C}})\times {\mathbb{C}}^*}(T^*G(v,N))$ by convolution is faithful.

Thus it is enough to check the relation in $\bigoplus K^{\operatorname {GL}_N({\mathbb{C}})\times {\mathbb{C}}^*}(T^*G(v,N))$.

Let ${\mathfrak{P}}^{(n)}(v,N)\subset {\mathfrak{M}}(v-n,N)\times {\mathfrak{M}}(v,N)$ be as in 5.3.1. It is the conormal bundle of

$$\begin{equation*} O^{(n)}(v,N) \overset{\operatorname {\scriptstyle def.}}{=}\{ (V^1, V^2) \in G(v-n, N)\times G(v, N) \mid V^1 \subset V^2 \}. \end{equation*}$$

We denote the projections for ${\mathfrak{P}}(v,N)$ by $p_1$, $p_2$, and the projections for $O(v,N)$ by $P_1$, $P_2$. Note that both $P_1$, $P_2$ are smooth and proper. Let $\pi _1$, $\pi _2$ denote the projections $T^*G(v-n,N)\to G(v-1,N)$, $T^*G(v,N)\to G(v,N)$.

Lemma 11.4.2 (58, Corollary 4).

For $E\in K^{\operatorname {GL}_N({\mathbb{C}})\times {\mathbb{C}}^*}(G(v,N))$ (resp. $E\in K^{\operatorname {GL}_N({\mathbb{C}})\times {\mathbb{C}}^*}(G(v-n,N))$), we have

$$\begin{align*} & [\mathcal{O}_{{\mathfrak{P}}^{(n)}(v,N)}] * \pi _2^* E = \sum _i (-q^{-2})^i \pi _1^* P_{1*} \left( [{\textstyle \bigwedge }^i TP_1] \otimes P_2^* E\right) \\ \Bigg (\text{resp. } & [\mathcal{O}_{{\mathfrak{P}}^{(n)}(v,N)}] * \pi _1^* E = \sum _i (-q^{-2})^i \pi _2^* P_{2*} \left( [{\textstyle \bigwedge }^i TP_2] \otimes P_1^* E\right)\Bigg ), \end{align*}$$

where $TP_1$ (resp. $TP_2$) is the relative tangent bundle along the fibers of $P_1$ (resp. $P_2$).

Proof.

As explained in 58, Corollary 4, the result follows from Lemma 6.3.1. The factor $q^{-2}$ is introduced to make the differential in the Koszul complex equivariant.

■

By the Thom isomorphism 13, 5.4.17,

$$\begin{equation*} \pi ^*\colon K^{\operatorname {GL}_N({\mathbb{C}})\times {\mathbb{C}}^*}(G(v,N))\to K^{\operatorname {GL}_N({\mathbb{C}})\times {\mathbb{C}}^*}(T^*G(v,N)) \end{equation*}$$

is an isomorphism. Moreover, we have the following explicit description of the $K$-group of the Grassmann variety (cf. 13, 6.1.6):

$$\begin{equation*} K^{\operatorname {GL}_N({\mathbb{C}})\times {\mathbb{C}}^*}(G(v,N)) \cong R({\mathbb{C}}^*\times \operatorname {GL}_v({\mathbb{C}})\times \operatorname {GL}_{N-v}({\mathbb{C}})) \cong {\mathbf{R}}^{S_{[v]}}, \end{equation*}$$

where $S_{[v]} = S_v\times S_{N-v}$ acts as permutations of $x_1$, …, $x_v$ and $x_{v+1}$, …, $x_N$. If $E$ denotes the tautological rank $v$ vector bundle over $G(v,N)$ and $Q$ denotes the quotient bundle $\mathcal{O}^{\oplus N}/E$, the isomorphism is given by

$$\begin{gather*} e_i(x_1,\dots , x_v) \mapsto {\textstyle \bigwedge }^i E, \qquad e_i(x_1^{-1},\dots , x_v^{-1}) \mapsto {\textstyle \bigwedge }^i E^*, \\ e_i(x_{v+1},\dots , x_N) \mapsto {\textstyle \bigwedge }^i Q, \quad e_i(x_{v+1}^{-1},\dots , x_N^{-1}) \mapsto {\textstyle \bigwedge }^i Q^*, \end{gather*}$$

where $e_i$ denotes the $i$th elementary symmetric polynomial.

The tautological vector bundle $V$ is isomorphic to $q E$, and $W$ is isomorphic to the trivial bundle $\mathcal{O}^{\oplus N}$. Let $C^\bullet (v,N)$ (resp. $C^{\prime \bullet }(v,N)$, $C^{\prime \prime \bullet }(v,N)$) be the complex 2.9.1 (resp. 9.3.1) over ${\mathfrak{M}}(v,N)$. In the description above, we have

$$\begin{equation} \begin{gathered} {\textstyle \bigwedge }_{-1/z} C^\bullet (v,N) = \left( \prod _{u\in [1,v]}(1 - z^{-1} q x_u) \right)^{-1} \prod _{t\in [v+1,N]} (1 - z^{-1} q^{-1} x_t), \\ \det C^{\prime \bullet }(v,N) = \prod _{t\in [v+1,N]} q^{-1} x_t, \quad \det C^{\prime \prime \bullet }(v,N) = \prod _{u\in [1,v]} q^{-1} x_u^{-1}. \end{gathered} {\tag{11.4.3}} \end{equation}$$

We also have

$$\begin{equation*} K^{\operatorname {GL}_N({\mathbb{C}})\times {\mathbb{C}}^*}(O^{(n)}(v,N)) \cong {\mathbf{R}}^{S_{[v-n]}\cap S_{[v]}}, \end{equation*}$$

where

$$\begin{equation*} S_{[v-n]}\cap S_{[v]} \cong S_{v-n}\times S_n\times S_{N-v} \end{equation*}$$

acts as permutations of $x_1$, …, $x_{v-n}$, $x_{v-n+1}$, …, $x_v$, and $x_{v+1}$, …, $x_N$. The natural vector bundle $V^2/V^1$ is $q (x_{v-n+1} + \cdots + x_{v})$. The relative tangent bundles $TP_1$, $TP_2$ are

$$\begin{equation*} [TP_1] = \sum _{t=v+1}^N \sum _{k=v-n+1}^v \frac{x_t}{x_k}, \qquad [TP_2] = \sum _{u=1}^{v-n} \sum _{k=v-n+1}^v \frac{x_k}{x_u}. \end{equation*}$$

Lemma 11.4.4 (58, Proposition 6).

(1) The pull-back homomorphisms

$$\begin{align*} &P_1^*\colon K^{\operatorname {GL}_N({\mathbb{C}})\times {\mathbb{C}}^*}(G(v-n,N))\to K^{\operatorname {GL}_N({\mathbb{C}})\times {\mathbb{C}}^*}(O(v,N)),\\ &P_2^*\colon K^{\operatorname {GL}_N({\mathbb{C}})\times {\mathbb{C}}^*}(G(v,N))\to K^{\operatorname {GL}_N({\mathbb{C}})\times {\mathbb{C}}^*}(O(v,N)) \end{align*}$$

are identified with the natural homomorphisms

$$\begin{equation*} {\mathbf{R}}^{S_{[v-n]}} \to {\mathbf{R}}^{S_{[v-n]}\cap S_{[v]}}, \quad {\mathbf{R}}^{S_{[v]}} \to {\mathbf{R}}^{S_{[v-n]}\cap S_{[v]}} \end{equation*}$$

respectively.

(2) The push-forward homomorphisms

$$\begin{align*} &P_{1*}\colon K^{\operatorname {GL}_N({\mathbb{C}})\times {\mathbb{C}}^*}(O(v,N))\to K^{\operatorname {GL}_N({\mathbb{C}})\times {\mathbb{C}}^*}(G(v-n,N)),\\ &P_{2*}\colon K^{\operatorname {GL}_N({\mathbb{C}})\times {\mathbb{C}}^*}(O(v,N))\to K^{\operatorname {GL}_N({\mathbb{C}})\times {\mathbb{C}}^*}(G(v,N)) \end{align*}$$

are identified with the natural homomorphisms

$$\begin{equation*} \begin{split} {\mathbf{R}}^{S_{[v-n]}\cap S_{[v]}} \ni f & \mapsto {\mathfrak{S}}_{[v]}^{[v-n]} \left(f \prod _{t=v+1}^{N}\prod _{k=v-n+1}^v ( 1 - \frac{x_k}{x_t})^{-1}\right), \\ {\mathbf{R}}^{S_{[v-n]}\cap S_{[v]}} \ni f & \mapsto {\mathfrak{S}}_{[v-n]}^{[v]} \left(f \prod _{u=1}^{v-1}\prod _{k=v-n+1}^v ( 1 - \frac{x_u}{x_k})^{-1}\right) \end{split} \end{equation*}$$

respectively. (The right hand sides are a priori in ${\mathcal{R}}$, but they are in fact in ${\mathbf{R}}$.)

Using the above lemmas, we can write the operators $x^+(z)$ explicitly as:

$$\begin{align*} x^+(z) f &= (-1)^{N-v} {\mathfrak{S}}_{[v]}^{[v-1]} \Bigg (f \sum _{r=-\infty }^\infty \left(\frac{x_v}{z}\right)^r x_v^{-v}\\ &\hphantom {= (-1)^{N-v} {\mathfrak{S}}_{[v]}^{[v-1]} \Bigg (f } \times \prod _{t\in [v+1,N]} q x_t^{-1} \left( 1 - \frac{x_v}{x_t}\right)^{-1} \left( 1 - q^{-2} \frac{x_t}{x_v}\right) \Bigg ) \\ &= \sum _{k\in [v,N]} f(x_{\tau _k^-[v-1]}) \sum _{r=-\infty }^\infty \left(\frac{x_k}{z}\right)^r x_k^{-N} \prod _{t\in [v, N]\setminus \{k\}} \frac{q x_k - q^{-1} x_t}{x_k - x_t}, \end{align*}$$

for $f\in {\mathbf{R}}^{S_{[v]}}$. Similarly,

$$\begin{align*} x^-(w) g &= (-1)^{1-v} {\mathfrak{S}}_{[v-1]}^{[v]} \Bigg (g \sum _{s=-\infty }^\infty \left(\frac{x_v}{w}\right)^s {x_v^{N-v+1}} \\ &\hphantom {= (-1)^{1-v} {\mathfrak{S}}_{[v-1]}^{[v]} \Bigg (g } \times \prod _{u\in [1,v-1]} q x_u \left( 1 - \frac{x_u}{x_v}\right)^{-1} \left( 1 - q^{-2} \frac{x_v}{x_u}\right) \Bigg ) \\ &= \sum _{l\in [1,v]} g(x_{\tau _l^+[v]}) \sum _{s=-\infty }^\infty \left(\frac{x_l}{w}\right)^s x_l^{N} \prod _{u\in [1,v]\setminus \{l\}} \frac{q^{-1} x_l - q x_u}{x_l - x_u} \end{align*}$$

for $g\in {\mathbf{R}}^{S_{[v-1]}}$.

Let us compare $x^+(z)x^+(w)$ with $x^+(w)x^+(z)$ in the component

$$\begin{equation*} K^{\operatorname {GL}_N({\mathbb{C}})\times {\mathbb{C}}^*}(T^*G(v,N)) \to K^{\operatorname {GL}_N({\mathbb{C}})\times {\mathbb{C}}^*}(T^*G(v-2,N)) \end{equation*}$$

as follows:

$$\begin{equation} \begin{split} & x^+(z)x^+(w) f\\ &\quad = - \sum _{l\in [v-1,N]}\sum _{k\in [v-1,N]\setminus \{l\}} f(x_{\tau _k^-\tau _l^-[v-2]}) \sum _{r=-\infty }^\infty \left(\frac{x_k}{w}\right)^r \sum _{s=-\infty }^\infty \left(\frac{x_l}{z}\right)^s x_k^{-N}x_l^{-N} \\ &\qquad \qquad \qquad \times \prod _{t\in [v-1, N]\setminus \{k,l\}} \frac{q x_k - q^{-1} x_t}{x_k - x_t} \prod _{u\in [v-1, N]\setminus \{l\}} \frac{q x_l - q^{-1} x_u}{x_l - x_u}, \\ & x^+(w)x^+(z) f\\ &\quad = - \sum _{k\in [v-1,N]}\sum _{l\in [v-1,N]\setminus \{k\}} f(x_{\tau _l^-\tau _k^-[v-2]}) \sum _{r=-\infty }^\infty \left(\frac{x_k}{w}\right)^r \sum _{s=-\infty }^\infty \left(\frac{x_l}{z}\right)^s x_k^{-N}x_l^{-N} \\ &\qquad \qquad \qquad \times \prod _{t\in [v-1, N]\setminus \{k\}} \frac{q x_k - q^{-1} x_t}{x_k - x_t} \prod _{u\in [v-1, N]\setminus \{k,l\}} \frac{q x_l - q^{-1} x_u}{x_l - x_u}. \end{split} {\tag{11.4.5}} \end{equation}$$

Hence we have

$$\begin{equation*} x^+(w) x^+(z) = \frac{q w - q^{-1}z}{q^{-1}w - qz} x^+(z) x^+(w). \end{equation*}$$

The relation 1.2.9 for $x^-(z)$, $x^-(w)$ can be proved in the same way.

Let us compare $x^-(w)x^+(z)$ and $x^+(z)x^-(w)$ in the component

$$\begin{equation*} K^{\operatorname {GL}_N({\mathbb{C}})\times {\mathbb{C}}^*}(T^*G(v,N)) \to K^{\operatorname {GL}_N({\mathbb{C}})\times {\mathbb{C}}^*}(T^*G(v,N)) \end{equation*}$$

as follows:

$$\begin{align*} x^-(w)x^+(z) f =& \sum _{l\in [1,v]} \sum _{k\in [v+1,N]\cup \{l\}} f(x_{\tau _k^-\tau _l^+[v]}) \sum _{r,s=-\infty }^\infty \left(\frac{x_k}{z}\right)^r \left(\frac{x_l}{w}\right)^s \left(\frac{x_l}{x_k}\right)^N \\ &\qquad \qquad \times \prod _{t\in ([v+1,N]\cup \{l\}) \setminus \{k\}} \frac{q x_k - q^{-1} x_t}{x_k - x_t} \prod _{u\in [1,v]\setminus \{l\}} \frac{q^{-1} x_l - q x_u}{x_l - x_u}, \\ x^+(z)x^-(w) f =& \sum _{k\in [v+1,N]}\sum _{l\in [1,v]\cup \{k\}} f(x_{\tau _l^+\tau _k^-[v]}) \sum _{r,s=-\infty }^\infty \left(\frac{x_k}{z}\right)^r \left(\frac{x_l}{w}\right)^s \left(\frac{x_l}{x_k}\right)^N \\ &\qquad \qquad \times \prod _{t\in [v+1,N] \setminus \{k\}} \frac{q x_k - q^{-1} x_t}{x_k - x_t} \prod _{u\in ([1,v]\cup \{k\}) \setminus \{l\}} \frac{q^{-1} x_l - q x_u}{x_l - x_u}. \end{align*}$$

Terms with $k\neq l$ cancel out for $x^+(z)x^-(w)f$ and $x^-(w)x^+(z)f$. Thus

$$\begin{align*} & \left[ x^+(z), x^-(w) \right]\\ &\quad = \sum _{k\in [v+1,N]} \sum _{r,s=-\infty }^\infty \left(\frac{x_k}{z}\right)^r \left(\frac{x_k}{w}\right)^s \prod _{t\in [v+1,N] \setminus \{k\}} \frac{q x_k - q^{-1} x_t}{x_k - x_t} \prod _{u\in [1,v]} \frac{q^{-1} x_k - q x_u}{x_k - x_u}\\ &\qquad \quad - \sum _{l\in [1,v]} \sum _{r,s=-\infty }^\infty \left(\frac{x_l}{z}\right)^r \left(\frac{x_l}{w}\right)^s \prod _{t\in [v+1,N]} \frac{q x_l - q^{-1} x_t}{x_l - x_t} \prod _{u\in [1,v]\setminus \{l\}} \frac{q^{-1} x_l - q x_u}{x_l - x_u}. \end{align*}$$

Let

$$\begin{equation*} \begin{split} A(x) & \overset{\operatorname {\scriptstyle def.}}{=}\prod _{u\in [1,v]} (x - x_u)\; \prod _{t\in [v+1,N]} (x - x_t), \\ B(x) & \overset{\operatorname {\scriptstyle def.}}{=}\prod _{u\in [1,v]} (q^{-1} x - q x_u)\; \prod _{t\in [v+1,N]} (q x - q^{-1} x_t). \end{split} \end{equation*}$$

Then we have

$$\begin{equation*} \left[ x^+(z), x^-(w)\right] = \frac{1}{q - q^{-1}}\sum _{m\in [1,N]} \sum _{r,s=-\infty }^\infty \left(\frac{x_m}{z}\right)^r \left(\frac{x_m}{w}\right)^s \frac{x_m^{-1}B(x_m)}{A'(x_m)}. \end{equation*}$$

Applying the residue theorem to $\frac{1}{q - q^{-1}} \sum _{r,s=-\infty }^\infty \left(\frac{x}{z}\right)^r \left(\frac{x}{w}\right)^s \frac{B(x)}{A(x)} \frac{dx}{x},$ we get

$$\begin{equation*} \left[ x^+(z), x^-(w)\right] = \frac{1}{q - q^{-1}}\left( \sum _{r=-\infty }^\infty \left(\frac{z}{w}\right)^r \left(\frac{B(z)}{A(z)}\right)^+ - \sum _{r=-\infty }^\infty \left(\frac{z}{w}\right)^r \left(\frac{B(z)}{A(z)}\right)^-\right), \end{equation*}$$

where $\left(\frac{B(z)}{A(z)}\right)^\pm \in {\mathbb{C}}[[z^{\mp }]]$ denotes the Laurent expansion of $\frac{B(z)}{A(z)}$ at $z = \infty$ and $0$ respectively. Since

$$\begin{equation*} \frac{B(z)}{A(z)} = q^{N-2v} \frac{{\textstyle \bigwedge }_{-1/(qz)}C^\bullet (v,N)}{{\textstyle \bigwedge }_{-q/z}C^\bullet (v,N)} \end{equation*}$$

by 11.4.3, we have completed the proof of Theorem 9.4.1.

12. Integral structure

In this section, we compare ${\mathbf{U}}^{{\mathbb{Z}}}_q({\mathbf{L}}{\mathfrak{g}})$ with $K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(Z({\mathbf{w}}))$. In the case of the affine Hecke algebra, the equivariant $K$-group of the Steinberg variety is isomorphic to the integral form of the affine Hecke algebra (see 13, 7.2.5). We shall prove a weaker form of the corresponding result for quiver varieties in this section.

12.1. Rank $1$ case

We first consider the case when the graph is of type $A_1$. We drop the subscript $k$. We use the notation in §11.4. We also consider $\omega ({\mathfrak{P}}^{(n)}(v,N))$ where ${\mathfrak{P}}^{(n)}(v,N)$ is as in 5.3.1 and $\omega \colon {\mathfrak{M}}(v-n,N)\times {\mathfrak{M}}(v,N)\to {\mathfrak{M}}(v,N)\times {\mathfrak{M}}(v-n,N)$ is the exchange of factors. We identify its equivariant $K$-group with ${\mathbf{R}}^{S_{[v-n]}\cap S_{[v]}}$ as in §11.4. In particular, the vector bundle $V^1/V^2$ is identified with $q(x_{v-n+1}+\cdots +x_v)$.

Lemma 12.1.1.

(1) Let $p_1 < \dots < p_s$ be an increasing sequence of integers and let $n_1$, …, $n_s$ be a sequence of positive integers such that $\sum n_i = n$. Let $\lambda$ be the partition

$$\begin{equation*} \left((p_2-p_1)^{n_2} \cdots (p_s-p_1)^{n_s}\right). \end{equation*}$$

Then for $g \in K^{\operatorname {GL}_N({\mathbb{C}})\times {\mathbb{C}}^*}(T^*G(v-n,N))$, we have

$$\begin{align*} & f_{p_1}^{(n_1)} f_{p_2}^{(n_2)} \cdots f_{p_s}^{(n_s)} g \\ &\qquad = \pm q^L \sum _{\{l_1,\dots ,l_n\}} g(x_{\tau _{l_1}^+\cdots \tau _{l_n}^+[v]}) {(x_{l_1}\cdots x_{l_n})}^{N+p_1} P_\lambda (x_{l_1},\dots , x_{l_n};q^2)\\ &\hphantom { = \pm q^L \sum _{\{l_1,\dots ,l_n\}} g(x_{\tau _{l_1}^+\cdots \tau _{l_n}^+[v]}) {(x_{l_1}\cdots x_{l_n})}} \times \prod _{\substack{i=1,\dots , n\\ u\in [1,v]\setminus \{l_1,\dots , l_n\}}} \frac{q^{-1} x_{l_i} - q x_u}{x_{l_i} - x_u} \end{align*}$$

for some $L\in {\mathbb{Z}}$. Here $P_\lambda$ is the Hall-Littlewood polynomial (see 41, III(2.1)), and the summation runs over the set of unordered $n$-tuples $\{ l_1,\dots ,l_n\} \subset [1,v]$ such that $l_i\neq l_j$ for $i\neq j$.

(2) Let us consider a tensor product $T(V^1/V^2)$ of exterior products of the bundle $V^1/V^2$ and its dual over $\omega ({\mathfrak{P}}^{(n)}(v,N))$, and denote by $T(x_{v-n+1}, \dots , x_v) \in {\mathbb{Z}}[x_{v-n+1}^{\pm },\dots , x_v^\pm ]^{S_n}\subset {\mathbf{R}}^{S_{[v-n]}\cap S_{[v]}}$ the corresponding element in the equivariant $K$-group. Then for $g\in {\mathbf{R}}^{S_{[v-n]}}$, we have the following formula:

$$\begin{align*} & \left[T(V^1/V^2)\otimes \det C^{\prime \prime \bullet }(v-n,N)^{\otimes -n}\right] \ast g \\ &\qquad = \pm \sum _{\{l_1,\dots ,l_n\}} g(x_{\tau _{l_1}^+\cdots \tau _{l_n}^+[v]})\, (x_{l_1}\dots x_{l_n})^{v-n}\, T(x_{l_1}, \dots , x_{l_n})\\ &\hphantom { = \pm \sum _{\{l_1,\dots ,l_n\}} g(x_{\tau _{l_1}^+\cdots \tau _{l_n}^+[v]})\, (x_{l_1}\dots x_{l_n})} \times \prod _{\substack{i=1,\dots , n\\ u\in [1,v]\setminus \{l_1,\dots , l_n\}}} \frac{q^{-1} x_{l_i} - q x_u}{x_{l_i} - x_u}, \end{align*}$$

where the summation runs over the set of unordered $n$-tuples $\{ l_1,\dots ,l_n\} \subset [1,v]$ such that $l_i\neq l_j$ for $i\neq j$.

Proof.

(1) Generalizing 11.4.5, we have the following formula for $f_{r_1}f_{r_2}\dots f_{r_n}\colon K^{\operatorname {GL}_N({\mathbb{C}})\times {\mathbb{C}}^*}(G(v-n,N)) \to K^{\operatorname {GL}_N({\mathbb{C}})\times {\mathbb{C}}^*}(G(v,N))$:

$$\begin{equation*} \begin{split} f_{r_1}f_{r_2}\dots f_{r_n} g =& \pm \sum _{(l_1, \dots , l_n)} g(x_{\tau _{l_1}^+\cdots \tau _{l_n}^+[v]}) x_{l_1}^{r_1}\cdots x_{l_n}^{r_n} {(x_{l_1}\cdots x_{l_n})}^{N} \\ &\qquad \qquad \times \prod _{\substack{i=1,\dots , n\\ t\in [1,v]\setminus \{l_1,\dots , l_n\}}} \frac{q^{-1} x_{l_i} - q x_u}{x_{l_i} - x_u} \prod _{i > j} \frac{q^{-1} x_{l_i} - q x_{l_j}}{x_{l_i} - x_{l_j}}, \end{split} \end{equation*}$$

where the summation runs over the set of ordered $n$-tuples $(l_1, \dots , l_n)$ such that $l_i\in [1,v]$, $l_i\neq l_j$ for $i\neq j$.

Choose $r_1 \le r_2 \le \dots \le r_n$ so that

$$\begin{equation*} (r_1,r_2,\dots ,r_n) = ( \underbrace{p_1,\dots ,p_1}_{\text{$n_1$ times}}, \underbrace{p_2,\dots ,p_2}_{\text{$n_2$ times}}, \dots ). \end{equation*}$$

Consider the following term which appeared in the above formula:

$$\begin{equation*} \begin{split} & \sum _{\sigma \in S_n} x_{l_{\sigma (1)}}^{r_1}\cdots x_{l_{\sigma (n)}}^{r_n} \prod _{i > j} \frac{q^{-1} x_{l_{\sigma (i)}} - q x_{l_{\sigma (j)}}}{x_{l_{\sigma (i)}} - x_{l_{\sigma (j)}}} \\ &\qquad = (x_{l_1}\cdots x_{l_n})^{r_1} \sum _{\sigma \in S_n} x_{l_{\sigma (1)}}^0x_{l_{\sigma (2)}}^{r_2-r_1}\cdots x_{l_{\sigma (n)}}^{r_n-r_1} \prod _{i > j} \frac{q^{-1} x_{l_{\sigma (i)}} - q x_{l_{\sigma (j)}}}{x_{l_{\sigma (i)}} - x_{l_{\sigma (j)}}}. \end{split} \end{equation*}$$

By 41, III(2.1) it is equal to

$$\begin{equation*} (x_{l_1}\cdots x_{l_n})^{r_1} q^{-n(n-1)/2} v_\lambda (q^2) P_\lambda (x_{l_1},\dots , x_{l_n};q^2), \end{equation*}$$

where $P_\lambda$ is the Hall-Littlewood polynomial and

$$\begin{equation*} v_\lambda (q^2) = q^{n_1(n_1-1)/2} [n_1]_q ! q^{n_2(n_2-1)/2} [n_2]_q ! \cdots q^{n_s(n_s-1)/2} [n_s]_q !\ . \end{equation*}$$

Thus we have the assertion.

(2) By Lemmas 11.4.2, 11.4.4 we have

$$\begin{align*} & \left[T(V^1/V^2)\otimes \det C^{\prime \prime \bullet }(v-n,N)^{\otimes -n}\right] \ast g \\ &\qquad = {\mathfrak{S}}_{[v-n]}^{[v]} \Bigg (g\, T(x_{v-n+1},\dots , x_v) \prod _{u\in [1,v-n]} (q x_u)^n \\ &\hphantom {= {\mathfrak{S}}_{[v-n]}^{[v]} \Big (g\, T(x_{v-n+1},\dots , x_v)} \times \prod _{l\in [v-n+1,v]} \left( 1 - \frac{x_u}{x_l}\right)^{-1} \left( 1 - q^{-2} \frac{x_l}{x_u}\right) \Bigg ) \\ &\qquad = \pm \sum _{\{l_1,\dots , l_n\}} g(x_{\tau _{l_1}^+\cdots \tau _{l_n}^+[v]})\, (x_{l_1}\dots x_{l_n})^{v-n}\, T(x_{l_1}, \dots , x_{l_n})\\ &\hphantom {= {\mathfrak{S}}_{[v-n]}^{[v]} \Big (g\, T(x_{v-n+1},\dots , x_v)} \times \prod _{\substack{i=1,\dots , n\\ u\in [1,v]\setminus \{l_1,\dots , l_n\}}} \frac{q^{-1} x_{l_i} - q x_u}{x_{l_i} - x_u}. \end{align*}$$ ■

12.2

Let $K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(Z({\mathbf{w}}))/ \operatorname {torsion}$ be

$$\begin{equation*} \operatorname {Image}\left(K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(Z({\mathbf{w}})) \to K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(Z({\mathbf{w}}))\otimes _{{\mathbb{Z}}[q,q^{-1}]}{\mathbb{Q}}(q) \right). \end{equation*}$$

(It seems reasonable to conjecture that $K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(Z({\mathbf{v}}^1,{\mathbf{v}}^2;{\mathbf{w}}))$ is free over $R(G_{\mathbf{w}}\times {\mathbb{C}}^*)$ since it is true for type $A_n$. But I do not know how to prove it in general.)

Theorem 12.2.1.

The homomorphism in Theorem 9.4.1 induces a homomorphism ${\mathbf{U}}^{{\mathbb{Z}}}_q({\mathbf{L}}{\mathfrak{g}})\to K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(Z({\mathbf{w}}))/ \operatorname {torsion}$.

Remark 12.2.2.

The homomorphism is neither injective nor surjective. It is likely that there exists a surjective homomorphism from a modification of ${\mathbf{U}}^{{\mathbb{Z}}}_q({\mathbf{L}}{\mathfrak{g}})$ to $K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(Z^{\operatorname {reg}}({\mathbf{w}}))/ \operatorname {torsion}$ for a suitable subset $Z^{\operatorname {reg}}({\mathbf{w}})$ of $Z({\mathbf{w}})$, as in 45, 9.5, 10.15.

Proof of Theorem 12.2.1.

It is enough to check that $e_{k,r}^{(n)}$, $f_{k,r}^{(n)}$, $q^h$ and the coefficients of $p_k^\pm (z)$ are mapped to $K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(Z({\mathbf{w}}))$. For $q^h$ and the coefficients of $p_k^\pm (z)$, the assertion is clear from the definition.

For $e_{k,r}^{(n)}$ and $f_{k,r}^{(n)}$, we can use a reduction to the rank $1$ case as in §11. Namely, it is enough to show the assertion when the graph is of type $A_1$.

Now if the graph is of type $A_1$, Lemma 12.1.1 together with Lemma 11.4.1 show that $f_{r}^{(n)}$ is represented by a certain line bundle over $\omega ({\mathfrak{P}}^{(n)}(v,N))$ extended to $Z(v,v-n;N)$ by $0$. We leave the proof for $e_r^{(n)}$ as an exercise. The only thing we need is to write down an analogue of Lemma 12.1.1 for $e_r^{(n)}$. It is straightforward.

■

12.3. The module $K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}({\mathfrak{L}}({\mathbf{w}}))$

By Theorems 12.2.1 and 7.3.5, $K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}({\mathfrak{L}}({\mathbf{w}}))$ is a ${\mathbf{U}}^{{\mathbb{Z}}}_q({\mathbf{L}}{\mathfrak{g}})$-module. We show that it is an l-highest weight module in this subsection.

Lemma 12.3.1.

Let ${\mathfrak{P}}_k^{(n)}({\mathbf{v}},{\mathbf{w}})$ be as in 5.3.1 and let $\omega \colon {\mathfrak{M}}({\mathbf{v}}-n\alpha _k,{\mathbf{w}})\times {\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})$ $\to {\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})\times {\mathfrak{M}}({\mathbf{v}}-n\alpha _k,{\mathbf{w}})$ denote the exchange of factors. Let $T(V^1_k/V^2_k)$ be a tensor product of exterior products of the vector bundle $V^1_k/V^2_k$ and its dual over $\omega ({\mathfrak{P}}_k^{(n)}({\mathbf{v}},{\mathbf{w}}))$. Let us consider it as an element of $K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(Z({\mathbf{v}},{\mathbf{v}}-n\alpha _k;{\mathbf{w}}))$. Then $\left[ T(V_k^1/V_k^2)\otimes \det C_k^{\prime \prime \bullet }({\mathbf{v}}-n\alpha _k,{\mathbf{w}})^{\otimes -n} \right]$ can be written as a linear combination (over ${\mathbb{Z}}[q,q^{-1}]$) of elements of the form

$$\begin{equation*} f_{k,p_1}^{(n_1)} f_{k,p_2}^{(n_2)} \cdots f_{k,p_s}^{(n_s)} \ast [\mathcal{O}_{\Delta {\mathfrak{M}}({\mathbf{v}}-n\alpha _k,{\mathbf{w}})}] \qquad (n_1+n_2+\cdots +n_s = n,\quad \text{$p_i$ distinct}), \end{equation*}$$

where $\Delta {\mathfrak{M}}({\mathbf{v}}-n\alpha _k,{\mathbf{w}})$ is the diagonal in ${\mathfrak{M}}({\mathbf{v}}-n\alpha _k,{\mathbf{w}})\times {\mathfrak{M}}({\mathbf{v}}-n\alpha _k,{\mathbf{w}})$.

Proof.

As in §11, we may assume that the graph is of type $A_1$. Now Lemma 12.1.1 together with the fact that Hall-Littlewood polynomials form a basis of symmetric polynomials implies the assertion.

■

Proposition 12.3.2.

Let $[0]\in K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}({\mathfrak{L}}(0,{\mathbf{w}}))$ be the class represented by the structure sheaf of ${\mathfrak{M}}(0,{\mathbf{w}})\cong {\mathfrak{L}}(0,{\mathbf{w}}) = \operatorname {point}$. Then

$$\begin{equation*} K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}({\mathfrak{L}}({\mathbf{w}})) = {\mathbf{U}}^{{\mathbb{Z}}}_q({\mathbf{L}}{\mathfrak{g}})^-\ast \left(R(G_{\mathbf{w}}\times {\mathbb{C}}^*)[0]\right). \end{equation*}$$

Proof.

The following proof is an adaptation of the proof of 45, 10.2, which was inspired by 35, 3.6 in turn.

We need the following notation:

$$\begin{equation*} {\mathfrak{L}}_{k;n}({\mathbf{v}},{\mathbf{w}}) \overset{\operatorname {\scriptstyle def.}}{=}{\mathfrak{L}}({\mathbf{v}},{\mathbf{w}})\cap {\mathfrak{M}}_{k;n}({\mathbf{v}},{\mathbf{w}}), \qquad {\mathfrak{L}}_{k;\le n}({\mathbf{v}},{\mathbf{w}}) \overset{\operatorname {\scriptstyle def.}}{=}{\mathfrak{L}}({\mathbf{v}},{\mathbf{w}})\cap {\mathfrak{M}}_{k;\le n}({\mathbf{v}},{\mathbf{w}}), \end{equation*}$$

$$\begin{equation*} {\mathfrak{L}}_{k;\ge n}({\mathbf{v}},{\mathbf{w}}) \overset{\operatorname {\scriptstyle def.}}{=}{\mathfrak{L}}({\mathbf{v}},{\mathbf{w}})\cap {\mathfrak{M}}_{k;\ge n}({\mathbf{v}},{\mathbf{w}}). \end{equation*}$$

We prove $K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}({\mathfrak{L}}({\mathbf{v}},{\mathbf{w}}))\subset {\mathbf{U}}^{{\mathbb{Z}}}_q({\mathbf{L}}{\mathfrak{g}})^-\ast \left(R(G_{\mathbf{w}}\times {\mathbb{C}}^*)[0]\right)$ by induction on the dimension vector ${\mathbf{v}}$. When ${\mathbf{v}}= 0$, the result is trivial since $K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(\operatorname {point}) = R(G_{\mathbf{w}}\times {\mathbb{C}}^*)$. Consider ${\mathfrak{L}}({\mathbf{v}},{\mathbf{w}})$ and suppose that

$$\begin{equation} \begin{split} &\text{if ${\mathbf{v}}- {\mathbf{v}}'\in \bigoplus {\mathbb{Z}}_{\ge 0}\alpha _k \setminus \{0\}$,}\\ &\text{then $K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}({\mathfrak{L}}({\mathbf{v}}',{\mathbf{w}}))\subset {\mathbf{U}}^{{\mathbb{Z}}}_q({\mathbf{L}}{\mathfrak{g}})^-\ast \left(R(G_{\mathbf{w}}\times {\mathbb{C}}^*)[0]\right)$.} \end{split}{\tag{12.3.3}} \end{equation}$$

Take $[E]\in K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}({\mathfrak{L}}({\mathbf{v}},{\mathbf{w}}))$. We want to show

$$\begin{equation*} [E]\in {\mathbf{U}}^{{\mathbb{Z}}}_q({\mathbf{L}}{\mathfrak{g}})^-\ast \left(R(G_{\mathbf{w}}\times {\mathbb{C}}^*)[0]\right). \end{equation*}$$

We may assume that the support of $E$ is contained in an irreducible component of ${\mathfrak{L}}({\mathbf{v}},{\mathbf{w}})$ without loss of generality. In fact, suppose that $\operatorname {Supp}E\subset X\cup Y$ such that $X$ is an irreducible component. Since $Y$ is a closed subvariety of $X\cup Y$ and since $X\cap Y$ is a closed subvariety of $X$, we have the diagram

$$\begin{equation*} \begin{CD} K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(Y) @>{i_*}>> K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(X\cup Y) @>{j^*}>> K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(X\setminus Y) @>>> 0 \\@AAA @A{i''_*}AA @| \\K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(X\cap Y) @>{i'_*}>> K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(X) @>{j^{\prime *}}>> K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(X\setminus Y) @>>> 0, \end{CD} \end{equation*}$$

where the first and the second row are exact by 6.1.2. Thus there exists $[E']\in K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(X)$ such that $j^{\prime *}[E'] = j^*[E]$. Then $j^*([E] - i''_*[E']) = 0$, and therefore there exists $E''\in K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(Y)$ such that $[E] = i_*[E''] + i''_*[E']$. By induction on the number of irreducible components in the support, we may assume that the support of $E$ is contained in an irreducible component, which is denoted by $X_E$.

Let us consider $\varepsilon _k$ defined in 2.9.3. If $\varepsilon _k(X_E) = 0$ for all $k\in I$, $X_E$ must be ${\mathfrak{L}}(0,{\mathbf{w}})$ by Lemma 2.9.4. We have nothing to prove in this case. Thus there exists $k$ such that $\varepsilon _k(X_E) > 0$. Set $n = \varepsilon _k(X_E)$. By the descending induction on $\varepsilon _k$, we may assume that

$$\begin{equation} \text{if $\operatorname {Supp}(E')\subset {\mathfrak{L}}_{k;\ge n+1}({\mathbf{v}},{\mathbf{w}})$, then $[E']\in {\mathbf{U}}^{{\mathbb{Z}}}_q({\mathbf{L}}{\mathfrak{g}})^-\ast \left(R(G_{\mathbf{w}}\times {\mathbb{C}}^*)[0]\right)$.} {\tag{12.3.4}} \end{equation}$$

Since ${\mathfrak{L}}_{k;\le n}({\mathbf{v}},{\mathbf{w}})$ is an open subvariety of ${\mathfrak{L}}({\mathbf{v}},{\mathbf{w}})$, we have an exact sequence

$$\begin{equation*} K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}({\mathfrak{L}}_{k;\ge n+1}({\mathbf{v}},{\mathbf{w}})) \xrightarrow {a_*} K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}({\mathfrak{L}}({\mathbf{v}},{\mathbf{w}})) \xrightarrow {b^*} K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}({\mathfrak{L}}_{k;\le n}({\mathbf{v}},{\mathbf{w}})) \to 0 \end{equation*}$$

by 6.1.2. Consider $b^*[E]$. By 12.3.4, it is enough to show that

$$\begin{equation} \text{there exists $[\widetilde{E}]\in {\mathbf{U}}^{{\mathbb{Z}}}_q({\mathbf{L}}{\mathfrak{g}})^-\ast [0]$ such that $b^*[E] = b^*[\widetilde{E}]$.} {\tag{12.3.5}} \end{equation}$$

Since $X_E\cap {\mathfrak{L}}_{k;\le n}({\mathbf{v}},{\mathbf{w}})\subset {\mathfrak{L}}_{k;n}({\mathbf{v}},{\mathbf{w}})$, the support of $b^*(E)$ is contained in ${\mathfrak{L}}_{k;n}({\mathbf{v}},{\mathbf{w}})$. We have a map

$$\begin{equation*} P\colon {\mathfrak{L}}_{k;n}({\mathbf{v}},{\mathbf{w}}) \to {\mathfrak{L}}_{k;0}({\mathbf{v}}-n\alpha _k,{\mathbf{w}}) \end{equation*}$$

which is the restriction of the map 5.4.2. Recall that this map is a Grassmann bundle (see Proposition 5.4.3). Let us denote its tautological bundle by $S$. Then $b^*[E]$ can be written as a linear combination of elements of the form

$$\begin{equation*} [T(S)]\otimes P^* [E_0], \end{equation*}$$

where $T(S)$ is a tensor product of exterior powers of the tautological bundle $S$ and $E_0\in K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}({\mathfrak{L}}_{k;0}({\mathbf{v}}-n\alpha _k,{\mathbf{w}}))$. Since the homomorphism

$$\begin{equation*} b^{\prime *}\colon K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}({\mathfrak{L}}({\mathbf{v}}-n\alpha _k,{\mathbf{w}}))\to K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}({\mathfrak{L}}_{k;0}({\mathbf{v}}-n\alpha _k,{\mathbf{w}})) \end{equation*}$$

is surjective by 6.1.2, there exists $[E_1]\in K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}({\mathfrak{L}}({\mathbf{v}}-n\alpha _k,{\mathbf{w}}))$ such that $b^{\prime *}[E_1] = [E_0]$.

Consider ${\mathfrak{P}}_k^{(n)\prime }({\mathbf{v}}-n\alpha _k,{\mathbf{w}})\cap ( {\mathfrak{L}}_{k;\le n}({\mathbf{v}},{\mathbf{w}})\times {\mathfrak{L}}({\mathbf{v}}-n\alpha _k,{\mathbf{w}}))$. By Proposition 5.4.3, it is isomorphic to ${\mathfrak{L}}_{k;n}({\mathbf{v}},{\mathbf{w}})$ and the map $P$ can be identified with the projection to the second factor. Moreover, the tautological bundle $S$ is identified with the restriction of the natural vector bundle $V_k^1/V_k^2$. Hence we have

$$\begin{equation*} [T(S)]\otimes P^* [E_0] = b^* \left( T(V_k^1/V_k^2) \ast [E_1]\right), \end{equation*}$$

where $T(V_k^1/V_k^2)$ is considered as an element of $K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(Z({\mathbf{v}},{\mathbf{v}}-n\alpha _k;{\mathbf{w}}))$. By Lemma 12.3.1, $T(V_k^1/V_k^2)$ can be written as a linear combination of elements

$$\begin{equation*} f_{k,p_1}^{(n_1)}f_{k,p_2}^{(n_2)}\dots f_{k,p_s}^{(n_s)}\ast [p_2^*\det C_k^{\prime \prime \bullet }({\mathbf{v}}-n\alpha _k,{\mathbf{w}})^{\otimes n}], \end{equation*}$$

where $p_2\colon \Delta {\mathfrak{M}}({\mathbf{v}}-n\alpha _k,{\mathbf{w}})\to {\mathfrak{M}}({\mathbf{v}}-n\alpha _k,{\mathbf{w}})$ is the projection. By 12.3.3,

$$\begin{equation*} [\det C_k^{\prime \prime \bullet }({\mathbf{v}}-n\alpha _k,{\mathbf{w}})^{\otimes n} \otimes E_1]\in {\mathbf{U}}^{{\mathbb{Z}}}_q({\mathbf{L}}{\mathfrak{g}})^-\ast \left(R(G_{\mathbf{w}}\times {\mathbb{C}}^*)[0]\right). \end{equation*}$$

Hence $T(V_k^1/V_k^2) \ast [E_1]\in {\mathbf{U}}^{{\mathbb{Z}}}_q({\mathbf{L}}{\mathfrak{g}})^-\ast \left(R(G_{\mathbf{w}}\times {\mathbb{C}}^*)[0]\right)$. Thus we have shown 12.3.5.

■

13. Standard modules

In this section, we start the study of the representation theory of ${\mathbf{U}}^{{\mathbb{Z}}}_q({\mathbf{L}}{\mathfrak{g}})$ using $K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(Z({\mathbf{w}}))$ and the homomorphism in 12.2.1. We shall define certain modules called standard modules, and study their properties. Results in this section hold even if $\varepsilon$ is a root of unity.

Note that $R(G_{\mathbf{w}}\times {\mathbb{C}}^*)$ is contained in the center of $K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(Z({\mathbf{w}}))$ by

$$\begin{equation*} R(G_{\mathbf{w}}\times {\mathbb{C}}^*)\ni \rho \mapsto \rho \otimes \sum _{\mathbf{v}}[\mathcal{O}_{\Delta {\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})}]. \end{equation*}$$

Hence a $K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(Z({\mathbf{w}}))/ \operatorname {torsion}$-module $M$ (over ${\mathbb{C}}$), which is l-integrable as a ${\mathbf{U}}^{{\mathbb{Z}}}_q({\mathbf{L}}{\mathfrak{g}})$-module, decomposes as $M = \bigoplus M_\chi$, where $\chi$ is a homomorphism from $R(G_{\mathbf{w}}\times {\mathbb{C}}^*)$ to ${\mathbb{C}}$ and $M_\chi$ is the corresponding simultaneous generalized eigenspace, i.e., some powers of the kernel of $\chi$ act as $0$ on $M_\chi$. Such a homomorphism $\chi$ is given by the evaluation of the character at a semisimple element $a = (s,\varepsilon )$ in $G_{\mathbf{w}}\times {\mathbb{C}}^*$. (This gives us a bijection between homomorphisms and semisimple elements.)

What is the meaning of the choice of $a = (s,\varepsilon )$ when we consider $M$ as a ${\mathbf{U}}^{{\mathbb{Z}}}_q({\mathbf{L}}{\mathfrak{g}})$-module? The role of $\varepsilon$ is clear. It is a specialization $q \to \varepsilon$, and we get ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})$-modules. It will become clear later that $s$ corresponds to the Drinfel’d polynomials by

$$\begin{equation*} P_k(u) = \text{(a normalization of) the characteristic polynomial of $s_k$}. \end{equation*}$$

13.1. Fixed data

Let $a = (s,\varepsilon )$ be a semisimple element in $G_{\mathbf{w}}\times {\mathbb{C}}^*$ and let $A$ be the Zariski closure of $\{ a^n \mid n\in {\mathbb{Z}}\}$. Let $\chi _a\colon R(A) \to {\mathbb{C}}$ be the homomorphism given by the evaluation at $a$. Considering ${\mathbb{C}}$ as an $R(A)$-module by this evaluation homomorphism, we denote it by ${\mathbb{C}}_a$. Via the homomorphism $R(G_{\mathbf{w}}\times {\mathbb{C}}^*)\to R(A)$, we consider ${\mathbb{C}}_a$ also as an $R(G_{\mathbf{w}}\times {\mathbb{C}}^*)$-module. We consider $R(A)$ as a ${\mathbb{Z}}[q,q^{-1}]$-algebra, where $R(G_{\mathbf{w}}\times {\mathbb{C}}^*)$ is a ${\mathbb{Z}}[q,q^{-1}]$-algebra as in §9.1.

Let ${\mathfrak{M}}({\mathbf{w}})^A$, ${\mathfrak{M}}_0(\infty ,{\mathbf{w}})^A$ be the fixed point subvarieties of ${\mathfrak{M}}({\mathbf{w}})$, ${\mathfrak{M}}_0(\infty ,{\mathbf{w}})$ respectively. Let us take a point $x\in {\mathfrak{M}}_0(\infty ,{\mathbf{w}})^A$ which is regular, i.e., $x\in {\mathfrak{M}}_0^{\operatorname {reg}}({\mathbf{v}}^0,{\mathbf{w}})$ for some ${\mathbf{v}}^0$.

The data $x$, $a$ will be fixed throughout this section.

13.2. Definition

As in 2.3.5, let ${\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})_x$ denote the inverse image of $x\in {\mathfrak{M}}_0^{\operatorname {reg}}({\mathbf{v}}^0,{\mathbf{w}})\subset {\mathfrak{M}}_0(\infty ,{\mathbf{w}})$ under the map $\pi \colon {\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})\to {\mathfrak{M}}_0({\mathbf{v}},{\mathbf{w}})\hookrightarrow {\mathfrak{M}}_0(\infty ,{\mathbf{w}})$. It is invariant under the $A$-action. Let ${\mathfrak{M}}({\mathbf{w}})_x$ be $\bigsqcup _{\mathbf{v}}{\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})_x$. We set

$$\begin{equation*} K^A({\mathfrak{M}}({\mathbf{w}})_x) \overset{\operatorname {\scriptstyle def.}}{=}\bigoplus _{\mathbf{v}}K^A({\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})_x) \end{equation*}$$

as convention.

Let $K^{A}(Z({\mathbf{w}}))/ \operatorname {torsion}$ be

$$\begin{equation*} \operatorname {Image}\left(K^{A}(Z({\mathbf{w}})) \to K^{A}(Z({\mathbf{w}}))\otimes _{{\mathbb{Z}}[q,q^{-1}]}{\mathbb{Q}}(q) \right). \end{equation*}$$

Let

$$\begin{equation} M_{x,a} \overset{\operatorname {\scriptstyle def.}}{=}K^A({\mathfrak{M}}({\mathbf{w}})_x)\otimes _{R(A)}{\mathbb{C}}_a. {\tag{13.2.1}} \end{equation}$$

By Theorem 7.3.5 together with Theorem 3.3.2, $K^A({\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})_x)$ is a free $R(A)$-module. Thus the $K^A(Z({\mathbf{w}}))$-module structure on $K^A({\mathfrak{M}}({\mathbf{w}})_x)$ descends to a $K^A(Z({\mathbf{w}}))/ \operatorname {torsion}$-module structure. Hence $M_{x,a}$ is a ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})$-module via the composition of

$$\begin{equation} \begin{split} {\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})& \to K^{G_{\mathbf{w}}\times {\mathbb{C}}^*}(Z({\mathbf{w}})) / \operatorname {torsion}\otimes _{R(G_{\mathbf{w}}\times {\mathbb{C}}^*)}{\mathbb{C}}_a \\ & \to K^{A}(Z({\mathbf{w}})) / \operatorname {torsion}\otimes _{R(A)}{\mathbb{C}}_a. \end{split} {\tag{13.2.2}} \end{equation}$$

We call $M_{x,a}$ the standard module.

It has a decomposition $M_{x,a} = \bigoplus K^A({\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})_x)\otimes _{R(A)}{\mathbb{C}}_a$, and each summand is a weight space:

$$\begin{equation} q^h\ast v = \varepsilon ^{\langle h, {\mathbf{w}}- {\mathbf{v}}\rangle } v \qquad \text{for $v\in K^A({\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})_x)\otimes _{R(A)}{\mathbb{C}}_a$}. {\tag{13.2.3}} \end{equation}$$

Thus $M_{x,a}$ has the weight decomposition as a ${\mathbf{U}}_{\varepsilon }({\mathfrak{g}})$-module.

In the remainder of this section, we study properties of $M_{x,a}$. The first one is the following.

Lemma 13.2.4.

As a ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})$-module, $M_{x,a}$ is l-integrable.

Proof.

The assertion is proved exactly as in 45, 9.3. Note that the regularity assumption of $x$ is not used here.

■

13.3. Highest weight vector

Recall that $\pi \colon {\mathfrak{M}}({\mathbf{v}}^0,{\mathbf{w}}) \to {\mathfrak{M}}_0({\mathbf{v}}^0,{\mathbf{w}})$ is an isomorphism on $\pi ^{-1}({\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}}^0,{\mathbf{w}}))$ (Proposition 2.6.2). Under this isomorphism, we can consider $x$ as a point in ${\mathfrak{M}}({\mathbf{v}}^0,{\mathbf{w}})$. Then ${\mathfrak{M}}({\mathbf{v}}^0,{\mathbf{w}})_x$ consists of the single point $x$, thus we have a canonical generator of $K^A({\mathfrak{M}}({\mathbf{v}}^0,{\mathbf{w}})_x)$. We denote it by $[x]$.

Since $x$ is fixed by $A$, the fibers $(V_k)_x$, $(W_k)_x$ of tautological bundles at $x$ are $A$-modules. Then the restriction of the complex $C_k^\bullet ({\mathbf{v}}^0,{\mathbf{w}})$ to $x$ can be considered as a complex of $A$-modules. In particular, it defines an element in $R(A)$. Let us denote it by $C_{k,x}^\bullet$.

Let us spell out $C_{k,x}^\bullet$ more explicitly. Since $x$ is fixed by $A$, we have a homomorphism $\rho \colon A\to G_{{\mathbf{v}}^0}$ by §4.1. It is uniquely determined by $x$ up to the conjugacy. Then a virtual $G_{{\mathbf{v}}^0}\times G_{\mathbf{w}}\times {\mathbb{C}}^*$-module

$$\begin{equation*} q^{-1} \left( \displaystyle {\bigoplus _{l}} [-\langle h_k,\alpha _l\rangle ]_q V_l^0 \oplus W_k\right) \end{equation*}$$

can be considered as a virtual $A$-module via $\rho \times (\operatorname {inclusion})\colon A\to G_{{\mathbf{v}}^0}\times G_{\mathbf{w}}\times {\mathbb{C}}^*$. Its isomorphism class is independent of $\rho$ and coincides with $C_{k,x}^\bullet$. Note that the first and third terms in 2.9.1 are absorbed in the term $l = k$.

Proposition 13.3.1.

The standard module $M_{x,a}$ is an l-highest weight module with l-highest weight $P_k(u) \overset{\operatorname {\scriptstyle def.}}{=}\chi _a\left({\textstyle \bigwedge }_{-u} C_{k,x}^\bullet \right)$. Namely, the following hold:

(1) $P_k(u)$ is a polynomial in $u$ of degree $\langle h_k, {\mathbf{w}}- {\mathbf{v}}^0\rangle$.

(2)

$$\begin{gather*} x_k^+(z)\ast [x] = 0, \quad q^h \ast [x] = \varepsilon ^{\langle h, {\mathbf{w}}- {\mathbf{v}}^0\rangle }[x], \\ p_k^+(z)\ast [x] = P_k(1/z)[x], \quad p_k^-(z)\ast [x] = (-z)^{\operatorname {rank}C_{k,x}^\bullet }P_k(1/z)\; \chi _{a} \left((\det C_{k,x}^{\bullet })^*\right) [x]. \end{gather*}$$

(3) $M_{x,a} = {\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})^-\ast [x].$

Proof.

(1) If we restrict the complex $C_k^\bullet ({\mathbf{v}}^0,{\mathbf{w}})$ to $x$, $\tau _k$ is surjective and $\sigma _k$ is injective by Lemma 2.9.2. Thus $C_{k,x}^\bullet$ is represented by a genuine $A$-module, and $\chi _a\left({\textstyle \bigwedge }_{-u} C_{k,x}^\bullet \right)$ is a polynomial in $u$. The degree is equal to $\langle h_k, {\mathbf{w}}- {\mathbf{v}}^0\rangle$ by the definition of $C_{k,x}$.

(2) The first equation is the consequence of ${\mathfrak{M}}({\mathbf{v}}-\alpha ^k,{\mathbf{w}})_x = \emptyset$, which follows from Lemma 2.9.4. The remaining equations follow from the definition and Lemma 8.1.1.

(3) The assertion is proved exactly as in Proposition 12.3.2. Note that the assumption $x\in {\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}}^0,{\mathbf{w}})$ is used here in order to apply Lemma 2.9.4.

■

Remark 13.3.2.

$P_k(u)$ is the Drinfel’d polynomial attached to the simple quotient of $M_{x,a}$, which we will study later.

We give a proof of Proposition 1.2.16 as promised:

Proof of Proposition 1.2.16.

It is enough to show that there exists a simple l-integrable l-highest module with given Drinfel’d polynomials $P_k(u)$. We can construct it as the quotient of the standard module $M_{0,a}$ by the unique maximal proper submodule. (The uniqueness can be proved as in the case of Verma modules.) Here the parameter $a = (s,\varepsilon )$ is chosen so that $P_k(u) = \chi _a\left({\textstyle \bigwedge }_{-u} q^{-1} W_k\right)$, i.e., $P_k(u)$ is a normalization of the characteristic polynomial of $s_k$.

■

13.4. Localization

Let $R(A)_a$ denote the localization of $R(A)$ with respect to $\operatorname {Ker}\chi _a$.

Let $Z({\mathbf{w}})^A$ denote the fixed point set of $A$ on $Z({\mathbf{w}})$, and let $i\colon {\mathfrak{M}}({\mathbf{w}})^A\times {\mathfrak{M}}({\mathbf{w}})^A \to {\mathfrak{M}}({\mathbf{w}})\times {\mathfrak{M}}({\mathbf{w}})$ be the inclusion. Note that it induces an inclusion $Z({\mathbf{w}})^A \to Z({\mathbf{w}})$ which we also denote by $i$. By the concentration theorem 53

$$\begin{equation*} i_* \colon K^A(Z({\mathbf{w}})^A)\otimes _{R(A)} R(A)_a \to K^A(Z({\mathbf{w}}))\otimes _{R(A)} R(A)_a \end{equation*}$$

is an isomorphism. Let

$$\begin{multline*} i^*\colon K^A(Z({\mathbf{w}})) \cong K^A({\mathfrak{M}}({\mathbf{w}})\times {\mathfrak{M}}({\mathbf{w}});Z({\mathbf{w}})) \\ \longrightarrow K^A({\mathfrak{M}}({\mathbf{w}})^A\times {\mathfrak{M}}({\mathbf{w}})^A;Z({\mathbf{w}})^A) \cong K^A(Z({\mathbf{w}})^A) \end{multline*}$$

be the pull-back with support map. Then $i^* i_*$ is given by multiplication by ${\textstyle \bigwedge }_{-1} N^*\boxtimes {\textstyle \bigwedge }_{-1} N^*$, where $N$ is the normal bundle of ${\mathfrak{M}}({\mathbf{w}})^A$ in ${\mathfrak{M}}({\mathbf{w}})$. By 13, 5.11.3, ${\textstyle \bigwedge }_{-1} N^*$ becomes invertible in the localized $K$-group. Thus $i^*$ is an isomorphism on the localized $K$-group. As in 13, 5.11.10, we introduce a correction factor to $i^*$:

$$\begin{equation*} r_a \overset{\operatorname {\scriptstyle def.}}{=}(1 \boxtimes ({\textstyle \bigwedge }_{-1} N^*)^{-1})\circ i^* \colon K^A(Z({\mathbf{w}}))\otimes _{R(A)} R(A)_a \to K^A(Z({\mathbf{w}})^A)\otimes _{R(A)} R(A)_a. \end{equation*}$$

Then $r_a$ is an algebra isomorphism with respect to the convolution.

Since $A$ acts trivially on $Z({\mathbf{w}})^A$, we have

$$\begin{equation} K^A(Z({\mathbf{w}})^A) \cong K(Z({\mathbf{w}})^A)\otimes R(A). {\tag{13.4.1}} \end{equation}$$

Thus we have the evaluation map

$$\begin{equation*} \operatorname {ev}_a\colon K^A(Z({\mathbf{w}})^A)\otimes _{R(A)} R(A)_a \cong K(Z({\mathbf{w}})^A)\otimes R(A)_a \to K(Z({\mathbf{w}})^A)\otimes {\mathbb{C}}, \end{equation*}$$

by sending $F\otimes (f/g)$ to $F\otimes (\chi _a(f)/\chi _a(g))$.

By the bivariant Riemann-Roch theorem 13, 5.11.11,

$$\begin{equation*} \operatorname {RR} \overset{\operatorname {\scriptstyle def.}}{=}(1\boxtimes \operatorname {Td}_{{\mathfrak{M}}({\mathbf{w}})^A})\cup \operatorname {ch}\colon K(Z({\mathbf{w}})^A)\to H_*(Z({\mathbf{w}})^A,{\mathbb{Q}}) \end{equation*}$$

is an algebra homomorphism with respect to the convolution. Here $\operatorname {ch}$ is the local Chern character homomorphism with respect to $Z({\mathbf{w}})^A\subset {\mathfrak{M}}({\mathbf{w}})^A\times {\mathfrak{M}}({\mathbf{w}})^A$ and $\operatorname {Td}_{{\mathfrak{M}}({\mathbf{w}})^A}$ is the Todd genus of ${\mathfrak{M}}({\mathbf{w}})^A$.

Composing 13.2.2 with all these homomorphisms, we have a homomorphism

$$\begin{equation} {\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})\to H_*(Z({\mathbf{w}})^A, {\mathbb{C}}). {\tag{13.4.2}} \end{equation}$$

Note that the torsion part in 13.2.2 disappears in the right hand side of 13.4.1 after tensoring with $R(A)_a$.

We have similar ${\mathbb{C}}$-linear maps for ${\mathfrak{M}}({\mathbf{w}})_x$:

$$\begin{equation} \begin{split} M_{x,a} = & \,K^A({\mathfrak{M}}({\mathbf{w}})_x)\otimes _{R(A)}{\mathbb{C}}_a \xrightarrow [\cong ]{i^*} K^A({\mathfrak{M}}({\mathbf{w}})_x^A)\otimes _{R(A)}{\mathbb{C}}_a \\ \xrightarrow [\cong ]{\operatorname {ev}_a} & K({\mathfrak{M}}({\mathbf{w}})_x^A)\otimes {\mathbb{C}}\xrightarrow [\cong ]{\operatorname {ch}} H_*({\mathfrak{M}}({\mathbf{w}})_x^A,{\mathbb{C}}), \end{split} {\tag{13.4.3}} \end{equation}$$

where $i^*$ is an isomorphism by the concentration theorem 53 and the invertibility of ${\textstyle \bigwedge }_{-1} N^*$ in the localized $K$-homology group, $\operatorname {ev}_a$ is an isomorphism since $A$ acts trivially on ${\mathfrak{M}}({\mathbf{w}})_x^A$, and $\operatorname {ch}$ is an isomorphism by Theorem 7.4.1 and Theorem 3.3.2. The composition is compatible with the ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})$-module structure, where $H_*({\mathfrak{M}}({\mathbf{w}})_x^A,{\mathbb{C}})$ is a ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})$-module via the convolution together with 13.4.2.

Recall that we have the decomposition

$$\begin{equation*} {\mathfrak{M}}({\mathbf{w}})^A = \bigsqcup _\rho {\mathfrak{M}}(\rho ), \end{equation*}$$

where $\rho$ runs over the set of homomorphisms $A\to G_{\mathbf{v}}$ (with various ${\mathbf{v}}$) (§4.1). Let

$$\begin{equation*} {\mathfrak{M}}(\rho )_x \overset{\operatorname {\scriptstyle def.}}{=}{\mathfrak{M}}(\rho )\cap {\mathfrak{M}}({\mathbf{w}})^A_x. \end{equation*}$$

Thus we have the canonical decomposition

$$\begin{equation} M_{x,a} = H_*({\mathfrak{M}}({\mathbf{w}})_x^A,{\mathbb{C}}) = \bigoplus _\rho H_*({\mathfrak{M}}(\rho )_x,{\mathbb{C}}). {\tag{13.4.4}} \end{equation}$$

Each summand $H_*({\mathfrak{M}}(\rho )_x,{\mathbb{C}})$ in 13.4.4 is an l-weight space with respect to the ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})$-action in the sense that operators $\psi _k^\pm (z)$ act on $H_*({\mathfrak{M}}(\rho )_x,{\mathbb{C}})$ as scalars plus nilpotent transformations. More precisely, we have

Proposition 13.4.5.

(1) Let $V_k$ be the tautological vector bundle over ${\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})$. Viewing ${\textstyle \bigwedge }_{u} V_k$ as an element of $K^A(\Delta {\mathfrak{M}}({\mathbf{v}},{\mathbf{w}}))[u]$, we consider it as an operator on $M_{x,a}$. Then we have

$$\begin{multline*} H_*({\mathfrak{M}}(\rho )_x,{\mathbb{C}}) = \{ m\in M_{x,a}\mid \left( {\textstyle \bigwedge }_{u} V_k - \chi _a({\textstyle \bigwedge }_{u} V_k)\operatorname {Id}\right)^N \ast m = 0 \\ \text{for $k\in I$ and sufficiently large $N$}\}, \end{multline*}$$

where $\chi _a({\textstyle \bigwedge }_{u} V_k)$ is the evaluation at $a$ of ${\textstyle \bigwedge }_{u} V_k$, considered as an $A$-module via $\rho \colon A\to G_{\mathbf{v}}$.

(2) Let us consider

$$\begin{equation*} C_k^\bullet ({\mathbf{v}},{\mathbf{w}}) = q^{-1} \left( \displaystyle {\bigoplus _{l}} [-\langle h_k,\alpha _l\rangle ]_q V_l \oplus W_k\right) \end{equation*}$$

as a virtual $A$-module via $\rho \times (\operatorname {inclusion})\colon A\to G_{\mathbf{v}}\times G_{\mathbf{w}}\times {\mathbb{C}}^*$. Then operators $\psi _k^\pm (z)$ act on $H_*({\mathfrak{M}}(\rho )_x,{\mathbb{C}})$ by

$$\begin{equation} \varepsilon ^{\operatorname {rank}C_k^\bullet ({\mathbf{v}},{\mathbf{w}})} \chi _a\left(\frac{{\textstyle \bigwedge }_{-1/qz}C_k^\bullet ({\mathbf{v}},{\mathbf{w}})}{{\textstyle \bigwedge }_{-q/z}C_k^\bullet ({\mathbf{v}},{\mathbf{w}})}\right)^\pm {\tag{13.4.6}} \end{equation}$$

plus nilpotent transformations.

Proof.

(2) follows from (1). We show (1).

Note that $\mathcal{O}_{\Delta {\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})}$ is mapped to $\mathcal{O}_{\Delta {\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})^A}$ under $r_a$, and $\mathcal{O}_{\Delta {\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})^A}$ is mapped to the fundamental class $[\Delta {\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})^A]$ under $\operatorname {RR}$. Combining with the projection formula 6.5.1, we find that the operator ${\textstyle \bigwedge }_u V_k$ is mapped to

$$\begin{equation*} \left(\operatorname {ch}\,\circ \, \operatorname {ev}_a\,\circ \, i^* {\textstyle \bigwedge }_u V_k\right) \cap [\Delta {\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})^A] \end{equation*}$$

under the homomorphism 13.4.2. Thus as an operator on $H_*({\mathfrak{M}}(\rho )_x, {\mathbb{C}})$, it is equal to

$$\begin{equation} m \mapsto (j^*\circ \operatorname {ch}\,\circ \, \operatorname {ev}_a\,\circ \, i^* {\textstyle \bigwedge }_u V_k)\cap m, {\tag{13.4.7}} \end{equation}$$

where $j\colon {\mathfrak{M}}(\rho )_x\to {\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})^A$ is the inclusion.

Now, on a connected space $X$, any $\alpha \in H^*(X,{\mathbb{C}})$ acts on $H_*(X,{\mathbb{C}})$ as a scalar plus nilpotent operator, where the scalar is the $H^0(X,{\mathbb{C}}) (\cong {\mathbb{C}})$-part of $\alpha$. In our situation, the $H^0$-part of 13.4.7 is given by $\chi _a({\textstyle \bigwedge }_u V_k)$. (Although we do not prove ${\mathfrak{M}}(\rho )_x$ is connected, the $H^0$-part is the same on any component.)

Furthermore, $\chi _a({\textstyle \bigwedge }_u V_k)$ determines all eigenvalues of the operator $a$ acting on $V_k$. Hence it determines the conjugacy class of the homomorphism $\rho \colon A\to G_{\mathbf{v}}$. Thus the generalized eigenspace of ${\textstyle \bigwedge }_u V_k$ with the eigenvalue $\chi _a({\textstyle \bigwedge }_u V_k)$ coincides with $H_*({\mathfrak{M}}(\rho )_x,{\mathbb{C}})$.

■

13.5. Frenkel-Reshetikhin’s $q$-character

In this subsection, we study Frenkel-Reshetikhin’s $q$-character for the standard module $M_{x,a}$. The result is a simple application of Proposition 13.4.5. Results in this subsection will not be used in the rest of the paper.

We assume ${\mathfrak{g}}$ is of type $ADE$ and $\varepsilon$ is not a root of unity in this subsection.

Let us recall the definition of $q$-character. It is a map from the Grothendieck group of finite dimensional ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})$-modules $M$. As we shall see later in §14.3, standard modules $M_{0,a}$ ($x = 0$ is fixed, ${\mathbf{w}}$ and $a = (s,\varepsilon )\in G_{\mathbf{w}}\times {\mathbb{C}}^*$ are moving) give a basis of the Grothendieck group, thus it is enough to define the $q$-character for standard modules $M_{0,a}$. We decompose $M = M_{0,a}$ as

$$\begin{equation*} M = \bigoplus M_{\Psi ^\pm }, \end{equation*}$$

as in 1.3.1. Moreover, by Proposition 13.4.5, $\Psi _k^\pm (z)$ have the form

$$\begin{equation} \Psi _k^\pm (z) = \varepsilon ^{\deg Q_k - \deg R_k} \frac{Q_k(1/\varepsilon z) R_k(\varepsilon /z)}{R_k(1/\varepsilon z)Q_k(\varepsilon /z)}, {\tag{13.5.1}} \end{equation}$$

where $Q_k(u)$, $R_k(u)$ are polynomials in $u$ with constant term $1$. (Compare with 18, Proposition 1. Note $u = 1/z$.) Suppose

$$\begin{equation*} \frac{Q_k(u)}{R_k(u)} = \frac{\prod _{r} (1 - u a_{kr})}{\prod _s (1 - u b_{ks})}. \end{equation*}$$

Then the $q$-character is defined by

$$\begin{equation*} \chi _q(M_{0,a}) \overset{\operatorname {\scriptstyle def.}}{=}\sum _{\Psi ^\pm (z)} \dim V_{\Psi ^\pm _k(z)}\, \prod _{k\in I}\prod _{r} Y_{k,a_{kr}} \prod _{s} Y_{k,b_{ks}}^{-1}, \end{equation*}$$

where $Y_{k,a_{kr}}$, $Y_{k,b_{ks}}$ are formal variables and $\chi _q$ takes its value in ${\mathbb{Z}}[Y_{k,c}^{\pm }]_{k\in I, c\in {\mathbb{C}}^*}$. ($\chi _q$ should not be confused with $\chi _a$.)

Let

$$\begin{equation*} A_{k,a} \overset{\operatorname {\scriptstyle def.}}{=}Y_{k,a\varepsilon } Y_{k,a\varepsilon ^{-1}} \prod _{h: \operatorname {in}(h) = k} Y_{\operatorname {out}(h), a\varepsilon ^{m(h)}}^{-1}. \end{equation*}$$

Proposition 13.5.2 (cf. Conjecture 1 in 18).

Let $M_{0,a}$ be a standard module with $x = 0$. Suppose that $P_k(u)$ in Proposition 13.3.1 equals

$$\begin{equation*} P_k(u) = \prod _{i=1}^{n_k} ( 1 - u a_i^{(k)}) \end{equation*}$$

for $k\in I$. Then the $q$-character of $M_{0,a}$ has the following form:

$$\begin{equation*} \prod _{k\in I} \prod _{i=1}^{n_k} Y_{k,a_i^{(k)}}\left(1+\sum M_p'\right), \end{equation*}$$

where each $M_p'$ is a product of $A_{l,c}^{-1}$ with $c\in \bigcup a_i^{(k)} \varepsilon ^{{\mathbb{Z}}}$.

Proof.

By Proposition 13.4.5, $H_*({\mathfrak{M}}(\rho )\cap \mathfrak{L}({\mathbf{w}}),{\mathbb{C}})$ is a generalized eigenspace for $\psi _k^\pm (z)$ for a homomorphism $\rho \colon A\to G_{\mathbf{v}}$. Thus it is enough to study the eigenvalue. We consider $V_k$, $W_k$ as $A$-modules via $\rho \times (\operatorname {inclusion})\colon A\to G_{\mathbf{v}}\times G_{\mathbf{w}}\times {\mathbb{C}}^*$ as before. Let $V_k(\lambda )$, $W_k(\lambda )$ be weight spaces as in §4.1.

By the definition of $C_k^\bullet ({\mathbf{v}},{\mathbf{w}})$ and $P_k(u)$, we have

$$\begin{equation*} \chi _a\left(\frac{{\textstyle \bigwedge }_{-1/qz}C_k^\bullet ({\mathbf{v}},{\mathbf{w}})}{{\textstyle \bigwedge }_{-q/z}C_k^\bullet ({\mathbf{v}},{\mathbf{w}})}\right) = \chi _a\left( \frac{{\textstyle \bigwedge }_{-1/qz}q^{-1}W_k}{{\textstyle \bigwedge }_{-q/z}q^{-1}W_k} \prod _l \frac{{\textstyle \bigwedge }_{-1/qz}q^{-1} [-\langle h_k, \alpha _l\rangle ]_q V_l}{{\textstyle \bigwedge }_{-q/z}q^{-1} [-\langle h_k, \alpha _l\rangle ]_q V_l}\right), \end{equation*}$$

$$\begin{equation*} P_k(u) = \chi _a\left({\textstyle \bigwedge }_{-u} q^{-1} W_k\right). \end{equation*}$$ By Proposition 13.4.5 we have

$$\begin{equation*} \frac{Q_k(u)}{R_k(u)} = P_k(u)\; \chi _a\left( \prod _l {\textstyle \bigwedge }_{-u}q^{-1} [-\langle h_k, \alpha _l\rangle ]_q V_l \right), \end{equation*}$$

where $Q_k(u)$, $R_k(u)$ are defined by 13.5.1.

Let $\{ c^{(l)}_t \}$ be the set of eigenvalues of $a\in A$ on $V_l$ counted with multiplicities. Then we have

$$\begin{equation*} \chi _a({\textstyle \bigwedge }_{-u} q^{-1} [-\langle h_k, \alpha _l\rangle ]_q V_l) = \begin{cases} {\displaystyle \left(\prod _t (1 - u c^{(k)}_t) (1 - u \varepsilon ^{2}c^{(k)}_t)\right)^{-1}} & \text{if $k=l$}, \\ {\displaystyle \prod _{h: \substack{\operatorname {in}(h) = k\\ \operatorname {out}(h) = l}} \prod _t (1 - u \varepsilon ^{m(h)+1} c^{(l)}_t)} & \text{otherwise}. \end{cases} \end{equation*}$$

Thus we have

$$\begin{equation*} \chi _q(M_{0,a}) = \sum _{\rho } \dim H_*({\mathfrak{M}}(\rho )\cap \mathfrak{L}({\mathbf{w}}),{\mathbb{C}}) \prod _{k\in I}\prod _{i=1}^{n_k} Y_{k,a_i^{(k)}} \prod _t A_{k,\varepsilon c^{(k)}_t}^{-1}. \end{equation*}$$

Note that the term for $\rho$ with ${\mathbf{v}}= 0$ has the contribution

$$\begin{equation*} \prod _{k\in I}\prod _{i=1}^{n_k} Y_{k,a_i^{(k)}}, \end{equation*}$$

and any other terms are monomials of $A_{k,\varepsilon c^{(k)}_t}^{-1}$ which are not constant.

Moreover, we have $c^{(k)}_t\in \bigcup a_i^{(k)} \varepsilon ^{{\mathbb{Z}}}$ by Lemma 4.1.4. This completes the proof.

■

14. Simple modules

The purpose of this section is to study simple modules of ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})$. Our discussion relies on Ginzburg’s classification of simple modules of the convolution algebra 13, Chapter 9. (See also 37.) He applied his classification to the affine Hecke algebra. However, unlike the case of the affine Hecke algebra, his classification does not directly imply a classification of simple modules of ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})$, and we need an extra argument. A difficulty lies in the fact that the homomorphism ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})\to H_*(Z({\mathbf{w}})^A,{\mathbb{C}})$ in 13.4.2 is not necessarily an isomorphism. Our additional input is Proposition 13.3.1(3). In order to illustrate its usage, we first consider the special case when $a = (s,\varepsilon )$ is generic in the first subsection. In this case, Ginzburg’s classification becomes trivial. Then we shall review Ginzburg’s classification in §14.2, and finally we shall study the general case in the last subsection.

We preserve the setup in §13.

14.1

Let us identify $e_{k,r}$, $f_{k,r}$ with their image under 13.4.2. Let $1_\rho \in H_*(Z({\mathbf{w}})^A)$ denote the fundamental class $[\Delta {\mathfrak{M}}(\rho )]$ of the diagonal of ${\mathfrak{M}}(\rho )\times {\mathfrak{M}}(\rho )$.

Lemma 14.1.1.

Let us consider

$$\begin{equation*} ({\mathfrak{M}}(\rho ^1)\times {\mathfrak{M}}(\rho ^2))\cap {\mathfrak{P}}_k({\mathbf{v}}^2,{\mathbf{w}})\subset ({\mathfrak{M}}({\mathbf{v}}^1,{\mathbf{w}})^A\times {\mathfrak{M}}({\mathbf{v}}^2,{\mathbf{w}})^A)\cap {\mathfrak{P}}_k({\mathbf{v}}^2,{\mathbf{w}}), \end{equation*}$$

and let $\lambda _0$ be the weight of $A$ determined by $\rho ^1$ and $\rho ^2$ as in §5.2. Then we have the following equality in $H_*(({\mathfrak{M}}(\rho ^1)\times {\mathfrak{M}}(\rho ^2))\cap {\mathfrak{P}}_k({\mathbf{v}}^2,{\mathbf{w}}),{\mathbb{C}})$:

$$\begin{align*} & p_2^* \operatorname {ch}\left({\textstyle \bigwedge }_{-u} V^2_l(\lambda )\right) \cap (1_{\rho ^1}x_k^+(z)1_{\rho ^2}) \\ &\quad = \begin{cases} p_1^*\operatorname {ch}\left({\textstyle \bigwedge }_{-u} V^1_l(\lambda )\right)\cap (1_{\rho ^1}x_k^+(z)1_{\rho ^2}) & \text{if $l\neq k$ or $\lambda \neq \lambda _0$}, \\ (1-\frac{uq}{z})p_1^*\operatorname {ch}\left({\textstyle \bigwedge }_{-u} V^1_l(\lambda )\right)\cap (1_{\rho ^1} x_k^+(z) 1_{\rho ^2}) & \text{if $l = k$, $\lambda = \lambda _0$}. \end{cases} \end{align*}$$

Proof.

We have the following equality in $K^0(({\mathfrak{M}}(\rho ^1)\times {\mathfrak{M}}(\rho ^2))\cap {\mathfrak{P}}_k({\mathbf{v}}^2,{\mathbf{w}}))$:

$$\begin{equation*} p_2^* V^2_l(\lambda ) = \begin{cases} p_1^* V^1_l(\lambda ) & \text{if $l\neq k$ or $\lambda \neq \lambda _0$}, \\ p_1^* V^1_k(\lambda _0) + (V^2/V^1) & \text{if $l = k$, $\lambda = \lambda _0$}, \end{cases} \end{equation*}$$

where $V^2/V^1$ is (the restriction of) the natural line bundle over ${\mathfrak{P}}_k({\mathbf{v}}^2,{\mathbf{w}})$. The assertion follows immediately.

■

Theorem 14.1.2.

Suppose that $a = (s,\varepsilon )$ is generic in the sense of Definition 4.2.1. (Hence, ${\mathfrak{L}}({\mathbf{w}})^A = {\mathfrak{M}}({\mathbf{w}})^A$.) Then the standard module

$$\begin{equation*} M_{0,a} = K^A({\mathfrak{M}}({\mathbf{w}})^A)\otimes _{R(A)}\otimes \,{\mathbb{C}}_a \cong H_*({\mathfrak{M}}({\mathbf{w}})^A,{\mathbb{C}}) \end{equation*}$$

is a simple ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})$-module. Its Drinfel’d polynomial is given by

$$\begin{equation*} P_k(u) = \det ( 1 - u\varepsilon ^{-1}s_k), \end{equation*}$$

where $s_k$ is the $\operatorname {GL}(W_k)$-component of $s\in G_{\mathbf{w}}$. Moreover, $M_{0,a}$ is isomorphic to a tensor product of l-fundamental representations when ${\mathfrak{g}}$ is finite dimensional.

Proof.

Recall that we have a distinguished vector (we denote it by $[0]$) in the standard module $M_{0,a}$ (§13.3). It has the properties listed in Proposition 13.3.1. In particular, it is the eigenvector for $p_k^\pm (z)$, and the eigenvalues are given in terms of $P_k(u)$ therein. In the present setting, $P_k(u)$ is equal to $\det ( 1 - u\varepsilon ^{-1}s_k)$.

Let

$$\begin{equation*} M_{0,a}^\circ \overset{\operatorname {\scriptstyle def.}}{=}\{ m\in M_{0,a} \mid \text{$e_{k,r}\ast m = 0$ for any $k\in I$, $r\in {\mathbb{Z}}$} \}. \end{equation*}$$

We have $[0]\in M_{0,a}^\circ$. We want to show that any nonzero submodule $M'$ of $M_{0,a}$ is $M_{0,a}$ itself. The weight space decomposition (as a ${\mathbf{U}}_{\varepsilon }({\mathfrak{g}})$-module) 13.2.3 of $M_{0,a}$ induces that of $M'$. Since the set of weights of $M'$ is bounded from ${\mathbf{w}}$ with respect to the dominance order, there exists a maximal weight of $M'$. Then a vector in the corresponding weight space is killed by all $e_{k,r}$ by the maximality. Thus $M'$ contains a nonzero vector $m\in M_{0,a}^\circ$. Hence it is enough to show that $M_{0,a}^\circ = {\mathbb{C}}[0]$ since we have already shown that $M_{0,a} = {\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})^-\ast [0]$ in Proposition 13.3.1(3).

Let us consider the operator

$$\begin{equation*} \left[ \Delta _* {\textstyle \bigwedge }_u V_l \right] \in K^A(Z({\mathbf{v}},{\mathbf{v}};{\mathbf{w}})) \end{equation*}$$

where $\Delta \colon {\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})\to Z({\mathbf{v}},{\mathbf{v}};{\mathbf{w}})$ is the diagonal embedding. If we consider such operators for various ${\mathbf{v}}$, $l\in I$, they form a commuting family. Moreover, $M_{0,a}^\circ$ is invariant under them since we have the relation

$$\begin{equation*} e_{k,r} \ast \left[ \Delta _* {\textstyle \bigwedge }_u V^2_l \right] = \begin{cases} \left[ \Delta _* {\textstyle \bigwedge }_u V^1_l \right] \ast e_{k,r} & \text{if $k\neq l$}, \\ \left[ \Delta _* {\textstyle \bigwedge }_u V^1_k \right] \ast (e_{k,r}+uq e_{k,r+1}) & \text{if $k = l$}, \end{cases} \end{equation*}$$

where $V^1_k$, $V^2_k$ are tautological bundles over ${\mathfrak{M}}({\mathbf{v}}^1,{\mathbf{w}})$, ${\mathfrak{M}}({\mathbf{v}}^2,{\mathbf{w}})$ respectively. (Here ${\mathbf{v}}^2 = {\mathbf{v}}^1 + \alpha _k$.) Thus $M_{0,a}^\circ$ is a direct sum of generalized eigenspaces for $\Delta _* {\textstyle \bigwedge }_u V_l$. Let us take a direct summand $M_{0,a}^{\circ \circ }$. By Proposition 13.4.5(1), $M_{0,a}^{\circ \circ }$ is contained in $H_*({\mathfrak{M}}(\rho ),{\mathbb{C}})$ for some $\rho \colon A\to G_{\mathbf{v}}$. If we can show $M_{0,a}^{\circ \circ } = {\mathbb{C}}[0]$, then we get $M_{0,a}^\circ = {\mathbb{C}}[0]$ since $M_{0,a}^{\circ \circ }$ is an arbitrary direct summand.

Since $a$ is generic, we have ${\mathfrak{M}}_0(\infty ,{\mathbf{w}})^A = \{0\}$. Hence

$$\begin{equation*} Z({\mathbf{w}})^A = {\mathfrak{M}}({\mathbf{w}})^A\times {\mathfrak{M}}({\mathbf{w}})^A, \end{equation*}$$

and ${\mathfrak{M}}({\mathbf{w}})^A$ is a nonsingular projective variety (having possibly infinitely many components). By the Poincaré duality, the intersection pairing

$$\begin{equation*} (\ ,\ )\colon H_*({\mathfrak{M}}({\mathbf{w}})^A,{\mathbb{C}})\otimes H_*({\mathfrak{M}}({\mathbf{w}})^A,{\mathbb{C}}) \to {\mathbb{C}} \end{equation*}$$

is nondegenerate.

Let $\,{}^{t}{f}_{k,r}$ denote the transpose of $f_{k,r}$ with respect to the pairing $(\ ,\ )$, namely

$$\begin{equation*} (f_{k,r}\ast m, m') = (m, \,{}^{t}{f}_{k,r}\ast m') \qquad \text{for $m$, $m'\in H_*({\mathfrak{M}}({\mathbf{w}})^A,{\mathbb{C}})$}. \end{equation*}$$

By the definition of the convolution, $\,{}^{t}{f}_{k,r}$ is equal to $\omega _* f_{k,r}$ where $\omega \colon {\mathfrak{M}}({\mathbf{w}})^A\times {\mathfrak{M}}({\mathbf{w}})^A \to {\mathfrak{M}}({\mathbf{w}})^A\times {\mathfrak{M}}({\mathbf{w}})^A$ is the map exchanging the first and second factors and $\omega _*$ is the induced homomorphism on $H_*({\mathfrak{M}}({\mathbf{w}})^A\times {\mathfrak{M}}({\mathbf{w}})^A,{\mathbb{C}})$.

Let us consider $1_\rho f_{k,r} 1_{\rho '}$, where $\rho$ is as above and $\rho '$ is any other homomorphism. It is just the projection of $f_{k,r}$ to the component $H_*({\mathfrak{M}}(\rho )\times {\mathfrak{M}}(\rho '),{\mathbb{C}})$. We have

$$\begin{equation*} 1_{\rho '}\,\,{}^{t}{f}_{k,r}\,1_{\rho } = 1_{\rho '}\, \omega _* f_{k,r}\, 1_{\rho } = (p_1^*\alpha \cup p_2^* \beta ) \cap 1_{\rho '}\, e_{k,r'}\, 1_{\rho } \end{equation*}$$

for some $r'\in {\mathbb{Z}}$, $\alpha \in H_*({\mathfrak{M}}(\rho ),{\mathbb{C}})$, and $\beta \in H_*({\mathfrak{M}}(\rho '),{\mathbb{C}})$. These $\alpha$ and $\beta$ come from asymmetry in the definition of $f_{k,r}$, $e_{k,r}$ and in the homomorphism 13.4.2. We do not give their explicit forms, although it is possible. What we need is for $\beta$ to be written by tensor powers of exterior products of $V_k^2(\lambda )$ for various $k$, $\lambda$. Thus we can write

$$\begin{equation*} (p_1^*\alpha \cup p_2^* \beta ) \cap 1_{\rho '}\, e_{k,r'}\, 1_{\rho } = \sum _{r''} p_1^* \alpha _{r''} \cap 1_{\rho '}\, e_{k,r''}\, 1_{\rho } \end{equation*}$$

for some $\alpha _{r''}\in H_*({\mathfrak{M}}(\rho ),{\mathbb{C}})$ by Lemma 14.1.1. Therefore, for $m\in M_{0,a}^{\circ \circ }$, we have

$$\begin{equation*} (f_{k,r}\ast m', m) = (1_\rho \, f_{k,r}\, 1_{\rho '}\ast m', m) = \left(m', \sum _{r''} p_1^* \alpha _{r''} \cap 1_{\rho '}\,e_{k,r''}\, 1_{\rho } \ast m\right) = 0 \end{equation*}$$

for any $k\in I$, $r\in {\mathbb{Z}}$, $\rho '$, $m'\in H_*({\mathfrak{M}}(\rho '),{\mathbb{C}})$. Here we have used $1_\rho \ast m= m$, $1_{\rho '}\ast m' = m'$, $e_{k,r''}\ast m = 0$. Since $H_*({\mathfrak{M}}({\mathbf{w}})^A,{\mathbb{C}}) = {\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})^-\ast [0]$, we have one of the following:

(a)

$(m',m) = 0$ for any $m'\in H_*({\mathfrak{M}}({\mathbf{w}})^A,{\mathbb{C}})$,

(b)

$m\in {\mathbb{C}}[0]$.

The first case is excluded by the nondegeneracy of $(\ ,\ )$. Thus we have $m \in {\mathbb{C}}[0]$.

Let us prove the last assertion. First consider the case ${\mathbf{w}}= \Lambda _k$ for some $k$. If $\varepsilon$ is not a root of unity, $a = (s,\varepsilon )\in {\mathbb{C}}^*\times {\mathbb{C}}^*$ is generic for the quiver variety ${\mathfrak{M}}(\Lambda _k)$. Hence the above shows that the standard module for ${\mathfrak{M}}(\Lambda _k)$ is simple, and hence gives an l-fundamental representation.

Let us return to the case for general ${\mathbf{w}}= \sum _k w_k \Lambda _k$. Let $a_k^{1}$, …, $a_k^{w_k}$ be eigenvalues of $s_k$ counted with multiplicities. By Proposition 1.2.19, it is enough to show that

$$\begin{equation} \dim M_{0,a} = \prod _k \prod _{i=1}^{w_k} \dim M_{0,a_k^i}(\Lambda _k), {\tag{14.1.3}} \end{equation}$$

where $M_{0,a_k}(\Lambda _k)$ is the standard module for ${\mathfrak{M}}(\Lambda _k)$ with $a_k = (s_k,\varepsilon )$. Since ${\mathfrak{M}}({\mathbf{w}})^A$ has no odd homology groups (Theorem 7.4.1), we have

$$\begin{equation*} \dim M_{0,a} = \operatorname {Euler}({\mathfrak{M}}({\mathbf{w}})^A), \end{equation*}$$

where $\operatorname {Euler}(\ )$ denotes the topological Euler number. By a property of the Euler number, we have

$$\begin{equation*} \operatorname {Euler}({\mathfrak{M}}({\mathbf{w}})^A) = \operatorname {Euler}({\mathfrak{M}}({\mathbf{w}})). \end{equation*}$$

If we take a maximal torus $T$ of $G_{\mathbf{w}}$, the fixed point set ${\mathfrak{M}}({\mathbf{w}})^T$ is isomorphic to $\prod _k {\mathfrak{M}}(\Lambda _k)^{w_k}.$ Again by a property of the Euler number, we have

$$\begin{equation*} \operatorname {Euler}({\mathfrak{M}}({\mathbf{w}})) = \prod _k \operatorname {Euler}({\mathfrak{M}}(\Lambda _k))^{w_k}. \end{equation*}$$

Since we have

$$\begin{equation*} \dim M_{0,a_k^i}(\Lambda _k) = \operatorname {Euler}({\mathfrak{M}}(\Lambda _k)), \end{equation*}$$

we get 14.1.3.

■

14.2. Simple modules of the convolution algebra

We briefly recall Ginzburg’s classification of simple modules of the convolution algebra 13, §8.6. (See also 37.)

Let $X$ be a complex algebraic variety. We consider the derived category of complexes of sheaves with constructible cohomology sheaves, and denote it by $D^b(X)$. We use the notation in 13. For example, we put

$$\begin{equation*} \operatorname {Ext}^k_{D^b(X)} (A, B) \overset{\operatorname {\scriptstyle def.}}{=}\operatorname {Hom}_{D^b(X)}(A, B[k]), \end{equation*}$$

$$\begin{equation*} \operatorname {Ext}^*_{D^b(X)}(A,B) \overset{\operatorname {\scriptstyle def.}}{=}\bigoplus _k \operatorname {Ext}^k_{D^b(X)}(A,B). \end{equation*}$$ $\operatorname {Ext}^*_{D^b(X)}(A,A)$ is an algebra by the Yoneda product. The Verdier duality operator is denoted by ${}^\vee$. Given graded vector spaces $V$, $W$, we write $V\overset{.}{=}W$ if there exists a linear isomorphism which does not necessarily preserve the gradings. We will also use the same notation to denote that two objects are quasi-isomorphic up to a shift in the derived category.

Let $f\colon M\to X$ be a projective morphism between algebraic varieties $M$, $X$, and assume that $M$ is nonsingular. Then we are in the setting for the convolution in §8 with $X_1 = X_2 = X_3$ and $Z_{12} = Z_{23} = Z$, where

$$\begin{equation*} Z \overset{\operatorname {\scriptstyle def.}}{=}M\times _X M = \{ (m^1, m^2)\in M\times M\mid f(m^1) = f(m^2) \}. \end{equation*}$$

Since $Z\circ Z = Z$, we have the convolution product

$$\begin{equation*} H_*(Z,{\mathbb{C}})\otimes H_*(Z,{\mathbb{C}}) \to H_*(Z,{\mathbb{C}}). \end{equation*}$$

Let $\mathcal{A}$ be the algebra $H_*(Z,{\mathbb{C}})$. Set $M_x = f^{-1}(x)$. Then the convolution defines an $\mathcal{A}$-module structure on $H_*(M_x,{\mathbb{C}})$. More generally, if $Y$ is a locally closed subset of $X$, then $H_*(f^{-1}(Y),{\mathbb{C}})$ has an $\mathcal{A}$-module structure via convolution.

By 13, 8.6.7, we have an algebra isomorphism, which does not necessarily preserve gradings,

$$\begin{equation*} \mathcal{A} = H_*(Z,{\mathbb{C}}) \overset{.}{=}\operatorname {Ext}^*_{D^b(X)}(f_*{\mathbb{C}}_M, f_*{\mathbb{C}}_M), \end{equation*}$$

where ${\mathbb{C}}_M$ is the constant sheaf on $M$.

We apply the decomposition theorem 6 to $f_*{\mathbb{C}}_M$. There exists an isomorphism in $D^b(X)$:

$$\begin{equation} f_*{\mathbb{C}}_M \cong \bigoplus _{\phi ,k} L_{\phi ,k}\otimes P_\phi [k], {\tag{14.2.1}} \end{equation}$$

where $\{ P_\phi \}$ is the set of isomorphism classes of simple perverse sheaves on $X$ such that some shift is a direct summand of $f_*{\mathbb{C}}_M$. We thus have an isomorphism

$$\begin{equation*} \mathcal{A} \overset{.}{=}\bigoplus _{i,j,k,\phi ,\psi } \operatorname {Hom}_{\mathbb{C}}(L_{\phi ,i}, L_{\psi ,j})\otimes \operatorname {Ext}^k_{D^b(X)}(P_\phi , P_\psi ). \end{equation*}$$

Set $L_\phi \overset{\operatorname {\scriptstyle def.}}{=}\bigoplus _k L_{\phi ,k}$ and

$$\begin{equation*} \mathcal{A}_k \overset{\operatorname {\scriptstyle def.}}{=}\bigoplus _{\phi ,\psi } \operatorname {Hom}_{\mathbb{C}}(L_{\phi }, L_{\psi })\otimes \operatorname {Ext}^k_{D^b(X)}(P_\phi , P_\psi ) \end{equation*}$$

so that $\mathcal{A} = \bigoplus \mathcal{A}_k$. By definition, $\mathcal{A}_k \cdot \mathcal{A}_l \subset \mathcal{A}_{k+l}$ under the multiplication of $\mathcal{A}$. By a property of perverse sheaves, we have $\mathcal{A}_k = 0$ for $k < 0$ and $\operatorname {Ext}^0_{D^b(X)}(P_\phi , P_\psi ) = {\mathbb{C}}\delta _{\phi \psi }\operatorname {id}$. Hence,

$$\begin{equation} \mathcal{A} = \mathcal{A}_0 \oplus \bigoplus _{k > 0} \mathcal{A}_k; \qquad \mathcal{A}_0 \cong \bigoplus _\phi \operatorname {End}(L_{\phi }). {\tag{14.2.2}} \end{equation}$$

In particular, the projection $\mathcal{A}\to \mathcal{A}_0$ is an algebra homomorphism. Furthermore, $\mathcal{A}_0$ is a semisimple algebra, and the kernel of the projection, i.e., $\bigoplus _{k > 0}\mathcal{A}_k$, consists of nilpotent elements, thus it is precisely the radical of $\mathcal{A}$. In particular,

$$\begin{equation*} \{ L_\phi \}_{\phi } \end{equation*}$$

is a complete set of mutually nonisomorphic simple $\mathcal{A}$-modules.

For $x\in X$, let $i_x\colon \{x\} \to X$ denote the inclusion. Then $H^*(i_x^! f_*{\mathbb{C}}_M)$ is an $\operatorname {Ext}^*_{D^b(X)}(f_*{\mathbb{C}}_M, f_*{\mathbb{C}}_M)$-module. More generally, if $i_Y\colon Y\hookrightarrow X$ is a locally closed embedding, then the hyper-cohomology groups $H^*(Y,i_Y^! f_*{\mathbb{C}}_M)$ and $H^*(Y,i_Y^*, f_*{\mathbb{C}}_M)$ are $\operatorname {Ext}^*_{D^b(X)}(f_*{\mathbb{C}}_M, f_*{\mathbb{C}}_M)$-modules. It is known (see 13, 8.6.16, 8.6.35) that $H^*(Y, i_Y^! f_*{\mathbb{C}}_M)$ is isomorphic to $H_*(f^{-1}(Y),{\mathbb{C}})$ as an $\mathcal{A} \cong \operatorname {Ext}^*_{D^b(X)}(f_*{\mathbb{C}}_M, f_*{\mathbb{C}}_M)$-module.

For $C\in D^b(\{x\})$, we write $H^k(C)$ instead of $H^k(\{x\}, C)$, and $H^*(C)$ instead of $H^*(\{x\}, C)$. By applying $H^*(i_x^!\bullet )$ to 14.2.1, we get an isomorphism

$$\begin{equation*} H_*(M_x,{\mathbb{C}}) \overset{.}{=}\bigoplus _{\phi ,k} L_{\phi }\otimes H^k(i_x^!P_\phi ). \end{equation*}$$

Let

$$\begin{equation*} M_{\ge k}(i_x^!f_*{\mathbb{C}}_M) \overset{\operatorname {\scriptstyle def.}}{=}\bigoplus _{k'\ge k} L_{\phi }\otimes H^{k'}(i_x^!P_\phi ). \end{equation*}$$

By definition, we have $\mathcal{A}_k\cdot M_{\ge l}(i_x^!f_*{\mathbb{C}}_M) \subset M_{\ge k+l}(i_x^!f_*{\mathbb{C}}_M)$ under the $\mathcal{A}$-module on $H^*(i_x^!f_*{\mathbb{C}}_M)$. In particular, $M_{\ge k}(i_x^!f_*{\mathbb{C}}_M)$ is an $\mathcal{A}$-submodule for each $k$. Hence

$$\begin{equation*} \operatorname {gr} M(i_x^!f_*{\mathbb{C}}_M) \overset{\operatorname {\scriptstyle def.}}{=}\bigoplus _k M_{\ge k}(i_x^!f_*{\mathbb{C}}_M)/ M_{\ge k+1}(i_x^!f_*{\mathbb{C}}_M) \end{equation*}$$

is an $\mathcal{A}$-module, on which $\bigoplus _{k > 0} \mathcal{A}_k$ acts as $0$. By definition,

$$\begin{equation*} \operatorname {gr} M(i_x^!f_*{\mathbb{C}}_M) \cong \bigoplus _\phi L_\phi \otimes H^*(i_x^!P_\phi ), \end{equation*}$$

where the $\mathcal{A}$-module structure on the right hand side is given by $a\colon \xi \otimes \xi '\mapsto a\xi \otimes \xi '$. Thus we have

Theorem 14.2.3.

In the Grothendieck group of $\mathcal{A}$-modules of finite dimension over ${\mathbb{C}}$, we have

$$\begin{equation*} H_*(M_x,{\mathbb{C}}) = \bigoplus _\phi L_\phi \otimes H^*(i_x^!P_\phi ), \end{equation*}$$

where the $\mathcal{A}$-module structure on the right hand side is given by $a\colon \xi \otimes \xi '\mapsto a\xi \otimes \xi '$.

Proof.

Since $\operatorname {gr} M(i_x^!f_*{\mathbb{C}}_M)$ is equal to $M(i_x^!f_*{\mathbb{C}}_M)$ in the Grothendieck group, the assertion follows from the discussion above.

■

14.3

In this subsection, we assume that the graph is of type $ADE$, and $\varepsilon$ is not a root of unity. We apply the results in the previous subsection to our quiver varieties.

Recall

$$\begin{equation*} Z({\mathbf{w}})^A = \{ (x^1, x^2)\in {\mathfrak{M}}({\mathbf{w}})^A\times {\mathfrak{M}}({\mathbf{w}})^A \mid \pi ^A(x^1) = \pi ^A(x^2) \}, \end{equation*}$$

where $\pi ^A\colon {\mathfrak{M}}({\mathbf{w}})^A\to {\mathfrak{M}}_0(\infty ,{\mathbf{w}})^A$ is the restriction of $\pi \colon {\mathfrak{M}}({\mathbf{w}})\to {\mathfrak{M}}_0(\infty ,{\mathbf{w}})$. Thus the results in the previous subsection are applicable to this setting. We have an algebra isomorphism

$$\begin{equation*} H_*(Z({\mathbf{w}})^A,{\mathbb{C}}) \overset{.}{=}\operatorname {Ext}^*_{D^b({\mathfrak{M}}_0(\infty ,{\mathbf{w}})^A)} (\pi ^A_*{\mathbb{C}}_{{\mathfrak{M}}({\mathbf{w}})^A}, \pi ^A_*{\mathbb{C}}_{{\mathfrak{M}}({\mathbf{w}})^A}). \end{equation*}$$

Let us denote this algebra by $\mathcal{A}$ as in the previous subsection.

Since the graph is of type $ADE$, we have ${\mathfrak{M}}_0(\infty ,{\mathbf{w}}) = \bigsqcup _{\mathbf{v}}{\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}})$ by Proposition 2.6.3. Thus we have the stratification ${\mathfrak{M}}_0(\infty ,{\mathbf{w}})^A = \bigsqcup {\mathfrak{M}_0^{\operatorname {reg}}}(\rho )$. (In fact, this holds under the same assumption as in Theorem 5.5.6: If we decompose $[B,i,j]\in {\mathfrak{M}}_0(\infty ,{\mathbf{w}})^A$ as in 45, 3.27, then the restriction of $B$ to $V^i$ for $i > 0$ is zero by Proposition 4.2.2 with ${\mathbf{w}}= 0$.) Since the restriction

$$\begin{equation*} \pi ^A|_{(\pi ^A)^{-1}({\mathfrak{M}_0^{\operatorname {reg}}}(\rho ))} \colon (\pi ^A)^{-1}({\mathfrak{M}_0^{\operatorname {reg}}}(\rho ))\to {\mathfrak{M}_0^{\operatorname {reg}}}(\rho ) \end{equation*}$$

is a locally trivial topological fibration by Theorem 3.3.2, all the complexes in the right hand side of 14.2.1 (applied to $f = \pi ^A$, $M = {\mathfrak{M}}({\mathbf{w}})^A$) have locally constant cohomology sheaves along each stratum ${\mathfrak{M}_0^{\operatorname {reg}}}(\rho )$. Since ${\mathfrak{M}_0^{\operatorname {reg}}}(\rho )$ is irreducible by Theorem 5.5.6, it implies that $P_\phi$ is the intersection cohomology complex $IC({\mathfrak{M}_0^{\operatorname {reg}}}(\rho ),\phi )$ associated with an irreducible local system $\phi$ on ${\mathfrak{M}_0^{\operatorname {reg}}}(\rho )$. Thus we have

$$\begin{equation} \pi ^A_*{\mathbb{C}}_{{\mathfrak{M}}({\mathbf{w}})^A} \cong \bigoplus _{(\rho ,\phi ,k)} L_{(\rho ,\phi ,k)}\otimes IC({\mathfrak{M}_0^{\operatorname {reg}}}(\rho ),\phi )[k] {\tag{14.3.1}} \end{equation}$$

for some finite dimensional vector space $L_{(\rho ,\phi ,k)}$. Let $L_{(\rho ,\phi )} \overset{\operatorname {\scriptstyle def.}}{=}\bigoplus _k L_{(\rho ,\phi ,k)}$. By a discussion in the previous subsection, $\{ L_{(\rho ,\phi )} \}$ is a complete set of mutually non-isomorphic simple $\mathcal{A}$-modules. Via the homomorphism 13.4.2, $L_{(\rho ,\phi )}$ is considered also as a ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})$-module.

Theorem 14.3.2.

Assume $\varepsilon$ is not a root of unity.

(1) Simple perverse sheaves $P_{\phi }$ whose shift appears in a direct summand of $\pi ^A_*{\mathbb{C}}_{{\mathfrak{M}}({\mathbf{w}})^A}$ are the intersection cohomology complexes $IC({\mathfrak{M}_0^{\operatorname {reg}}}(\rho ))$ associated with the constant local system ${\mathbb{C}}_{{\mathfrak{M}_0^{\operatorname {reg}}}(\rho )}$ on various ${\mathfrak{M}_0^{\operatorname {reg}}}(\rho )$.

(2) Let us denote the constant local system ${\mathbb{C}}_{{\mathfrak{M}_0^{\operatorname {reg}}}(\rho )}$ by ${\mathbb{C}}_\rho$ for simplicity. Then $L_{(\rho ,{\mathbb{C}}_\rho )}$ is nonzero if and only if ${\mathfrak{M}_0^{\operatorname {reg}}}(\rho )\neq \emptyset$. Moreover, there is a bijection between the set $\{\rho \mid L_{(\rho ,{\mathbb{C}}_{\rho })}\neq 0\}$ and the set of l-weights of $M_{0,a}$ which are l-dominant.

(3) The simple $\mathcal{A} = H_*(Z({\mathbf{w}})^A,{\mathbb{C}})$-module $L_{(\rho ,{\mathbb{C}}_\rho )} = \bigoplus _k L_{(\rho ,{\mathbb{C}}_\rho ,k)}$ is also simple as a ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})$-module, and its Drinfel’d polynomial is $P_k(u) = \chi _a\left({\textstyle \bigwedge }_{-u} C_{k,x}^\bullet \right)$ in Proposition 13.3.1 for $x\in {\mathfrak{M}_0^{\operatorname {reg}}}(\rho )$.

(4) $L_{(\rho ,{\mathbb{C}}_\rho )}$ is the simple quotient of $M_{x,a}$, where $x$ is a point in a stratum ${\mathfrak{M}_0^{\operatorname {reg}}}(\rho )$.

(5) Standard modules $M_{x,a}$ and $M_{y,a}$ are isomorphic as ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})$-modules if and only if $x$ and $y$ are contained in the same stratum.

Proof.

We use the transversal slice in §3.3. The idea to use transversal slices is taken from 13, §8.5.

Choose and fix a point $x\in {\mathfrak{M}}_0(\infty ,{\mathbf{w}})^A$. Suppose that $x$ is contained in a stratum ${\mathfrak{M}_0^{\operatorname {reg}}}(\rho _x)$ for some $\rho _x$. We first show

Claim.

If ${\mathbb{C}}_{\rho _x}$ denotes the constant local system on ${\mathfrak{M}_0^{\operatorname {reg}}}(\rho _x)$, the corresponding vector space $L_{(\rho _x,{\mathbb{C}}_{\rho _x})}$ is nonzero.

If we restrict $\pi ^A$ to the component ${\mathfrak{M}}(\rho _x)$, then we have

$$\begin{equation} \pi ^A_*{\mathbb{C}}_{{\mathfrak{M}}(\rho _x)} \overset{.}{=}\bigoplus _{(\rho ,\phi )} L_{(\rho ,\phi )}'\otimes IC({\mathfrak{M}_0^{\operatorname {reg}}}(\rho ),\phi ), {\tag{14.3.3}} \end{equation}$$

where $L_{(\rho ,\phi )}'$ is a direct summand of $L_{(\rho ,\phi )}$. The summation runs over the set of pairs $(\rho ,\phi )$ such that ${\mathfrak{M}_0^{\operatorname {reg}}}(\rho )$ is contained in $\pi ^A({\mathfrak{M}}(\rho _x))$. (In fact, 14.3.1 was obtained by applying the decomposition theorem to each component ${\mathfrak{M}}(\rho _x)$ and taking the direct sum.) If we restrict 14.3.3 to the open stratum ${\mathfrak{M}_0^{\operatorname {reg}}}(\rho _x)$ of $\pi ^A({\mathfrak{M}}(\rho _x))$, the right hand side of 14.3.3 becomes

$$\begin{equation*} \bigoplus _\phi L_{(\rho _x,\phi )}'\otimes \phi , \end{equation*}$$

where the summation runs over the set of isomorphism classes of irreducible local systems $\phi$ on ${\mathfrak{M}_0^{\operatorname {reg}}}(\rho _x)$. On the other hand, $\pi ^A$ induces an isomorphism between $(\pi ^A)^{-1}({\mathfrak{M}_0^{\operatorname {reg}}}(\rho _x))$ and ${\mathfrak{M}_0^{\operatorname {reg}}}(\rho _x)$ by Proposition 2.6.2. This means that the restriction of the left hand side of 14.3.3 is the constant local system ${\mathbb{C}}_{\rho _x}$. Hence we have $L_{(\rho _x,{\mathbb{C}}_{\rho _x})}' \cong {\mathbb{C}}$, and $L_{(\rho _x,{\mathbb{C}}_{\rho _x})}$ is nonzero. This is the end of the proof of the claim.

The claim implies the first assertion of (2). Let us prove the latter assertion of (2). Suppose ${\mathfrak{M}_0^{\operatorname {reg}}}(\rho )\neq \emptyset$. Then we have ${\mathfrak{M}}(\rho )\neq \emptyset$ and $H_*({\mathfrak{M}}(\rho )\cap {\mathfrak{L}}({\mathbf{w}}),{\mathbb{C}})\neq 0$ by Proposition 4.1.2. By Proposition 13.4.5, the corresponding l-weight space is nonzero, where the l-weight $\Psi ^\pm (z) = (\Psi _k^\pm (z))_k$ is given by 13.4.6. Furthermore, since $C_k^\bullet ({\mathbf{v}},{\mathbf{w}})$ can be represented by a genuine $A$-module over a point in ${\mathfrak{M}_0^{\operatorname {reg}}}(\rho )$ by Lemma 2.9.2, $\chi _a\left({\textstyle \bigwedge }_{-u} C_k^\bullet ({\mathbf{v}},{\mathbf{w}})\right)$ is a polynomial in $u$. Thus $\Psi ^\pm (z)$ is l-dominant.

Conversely suppose that we have the l-weight space with the l-weight 13.4.6 nonzero. Since $\varepsilon$ is not a root of unity, the $\varepsilon$-analogue of the Cartan matrix $[-\langle h_k,\alpha _l\rangle ]_\varepsilon$ is invertible. Hence 13.4.6 determines $\chi _a({\textstyle \bigwedge }_{u} V_k)$, and a homomorphism $\rho \colon A\to G_{\mathbf{v}}$. Moreover, the l-weight space is precisely $H_*({\mathfrak{M}}(\rho )\cap {\mathfrak{L}}({\mathbf{w}}),{\mathbb{C}})$ by Proposition 13.4.5(1). In particular, we have $H_*({\mathfrak{M}}(\rho )\cap {\mathfrak{L}}({\mathbf{w}}),{\mathbb{C}})\neq 0$, and hence ${\mathfrak{M}}(\rho )\neq \emptyset$. Furthermore, if we decompose $C_k^\bullet ({\mathbf{v}},{\mathbf{w}})$ into $\bigoplus _\lambda C_{k,\lambda }^\bullet (\rho )$ as in §4.1, we have $\operatorname {rank}C_{k,\lambda }^\bullet (\rho ) \ge 0$ since the l-weight 13.4.6 is l-dominant. By Corollary 5.5.5, $\tau _{k,\lambda }$ is surjective for any $k$, $\lambda$ on a nonempty open subset of ${\mathfrak{M}}(\rho )$. By Lemma 2.9.4, ${\mathfrak{M}_0^{\operatorname {reg}}}(\rho )$ is nonempty. This shows the latter half of (2).

Let ${\mathbf{v}}_x$ denote the dimension vector corresponding to $\rho _x$, i.e., ${\mathfrak{M}_0^{\operatorname {reg}}}(\rho _x)\subset {\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}}_x,{\mathbf{w}})$. Take a transversal slice to ${\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}}_x,{\mathbf{w}})$ at $x$ as in §3.3. Let $S$ be its intersection with ${\mathfrak{M}}_0(\infty ,{\mathbf{w}})^A$. Since the transversal slice in §3.3 can be made $A$-equivariant (Remark 3.3.3), it is a transversal slice to ${\mathfrak{M}_0^{\operatorname {reg}}}(\rho _x)$ (at $x$) in ${\mathfrak{M}}_0(\infty ,{\mathbf{w}})^A$. Let $\widetilde{S} \overset{\operatorname {\scriptstyle def.}}{=}(\pi ^A)^{-1}(S)$. Let $\varepsilon \colon S\to {\mathfrak{M}}_0(\infty ,{\mathbf{w}})^A$, $\widetilde{\varepsilon }\colon \widetilde{S}\to {\mathfrak{M}}({\mathbf{w}})^A$ be the inclusions.

The stratification ${\mathfrak{M}}_0({\mathbf{w}})^A = \bigsqcup {\mathfrak{M}_0^{\operatorname {reg}}}(\rho )$ induces by restriction a stratification $S = \bigsqcup S_\rho$ where $S_\rho = {\mathfrak{M}_0^{\operatorname {reg}}}(\rho )\cap S$. Any intersection complex $IC({\mathfrak{M}_0^{\operatorname {reg}}}(\rho ),\phi )$ restricts (up to shift) to the intersection complex $IC(S_\rho ,\phi |_{S_\rho })$ by transversality. Here $\phi |_{S_\rho }$ is the restriction of $\phi$ to $S_{\rho }$. Taking $\varepsilon ^!$ of 14.3.1, we get

$$\begin{equation} \varepsilon ^! \left(\pi ^A_*{\mathbb{C}}_{{\mathfrak{M}}({\mathbf{w}})^A}\right) \overset{.}{=}\bigoplus _{(\rho ,\phi )} L_{(\rho ,\phi )}\otimes IC(S_\rho ,\phi |_{S_\rho }). {\tag{14.3.4}} \end{equation}$$

Let $i^S_x\colon \{x\}\to S$ be the inclusion. It induces two pull-back homomorphisms $i^{S!}_x$, $i^{S*}_x$, and there is a natural morphism $i^{S!}_x E \to i^{S*}_x E$ for any $E\in D^b(S)$. We apply these functors to both sides of 14.3.4 and take cohomology groups. By a property of intersection cohomology sheaves (see 13, 8.5.3), the homomorphism

$$\begin{equation} H^*(i^{S!}_x IC(S_\rho ,\phi |_{S_\rho })) \to H^*(i^{S*}_x IC(S_\rho ,\phi |_{S_\rho })) {\tag{14.3.5}} \end{equation}$$

is zero unless $S_\rho = \{x\}$ (or equivalently $\rho = \rho _x$), in which case it is a quasi-isomorphism. Thus

$$\begin{equation} \operatorname {Im}\left[ H^*(i^{S!}_x \varepsilon ^! \pi ^A_* {\mathbb{C}}_{{\mathfrak{M}}({\mathbf{w}})^A}) \to H^*(i^{S*}_x \varepsilon ^! \pi ^A_* {\mathbb{C}}_{{\mathfrak{M}}({\mathbf{w}})^A}) \right] \overset{.}{=}\bigoplus _{\phi } L_{(\rho _x,\phi )}\otimes \phi _{x}, {\tag{14.3.6}} \end{equation}$$

where the summation runs over isomorphism classes of irreducible local systems on ${\mathfrak{M}_0^{\operatorname {reg}}}(\rho _x)$, and $\phi _{x}$ is the fiber of the local system $\phi$ at $x$. Moreover, 14.3.5 is a homomorphism of $\mathcal{A}$-modules, and 14.3.6 is an isomorphism of $\mathcal{A}$-modules, where the module structure on the right hand side is given by $a\colon \xi \otimes \xi '\mapsto a\xi \otimes \xi '$.

On the other hand, we have

$$\begin{equation*} H^*(i^{S!}_x \varepsilon ^! \pi ^A_* {\mathbb{C}}_{{\mathfrak{M}}({\mathbf{w}})^A}) = H^*(i_x^! \pi ^A_* {\mathbb{C}}_{\widetilde{S}}) \overset{.}{=}H_*({\mathfrak{M}}({\mathbf{w}})^A_x,{\mathbb{C}}). \end{equation*}$$

As shown in 13.4.3, the right hand side is isomorphic to the standard module $M_{x,a}$. Thus the left hand side of 14.3.6 is a quotient of $M_{x,a}$, and it is indecomposable by Proposition 13.3.1(3). Thus the right hand side of 14.3.6 consists of at most a single direct summand. Since we have already shown that $L_{(\rho _x,{\mathbb{C}}_{\rho _x})}\neq 0$ in the claim, we get $L_{(\rho _x,\phi )} = 0$ if $\phi$ is a nonconstant irreducible local system. Since $x$ was an arbitrary point, we have the statement (1).

Let us prove (3). For the proof, we need a further study of 14.3.6. By the above discussion, we have

$$\begin{equation} \operatorname {Im}\left[ H^*(i^{S!}_x \varepsilon ^! \pi ^A_* {\mathbb{C}}_{{\mathfrak{M}}({\mathbf{w}})^A}) \to H^*(i^{S*}_x \varepsilon ^! \pi ^A_* {\mathbb{C}}_{{\mathfrak{M}}({\mathbf{w}})^A}) \right] \overset{.}{=}L_{(\rho _x,{\mathbb{C}}_{\rho _x})}. {\tag{14.3.7}} \end{equation}$$

By the base change theorem, we have $\varepsilon ^! \left(\pi ^A_*{\mathbb{C}}_{{\mathfrak{M}}({\mathbf{w}})^A}\right) = \pi ^S_* {\widetilde{\varepsilon }\,}^!{\mathbb{C}}_{{\mathfrak{M}}({\mathbf{w}})^A}$ where $\pi ^S$ is the restriction of $\pi ^A$ to $\widetilde{S}$. Further, we have ${\widetilde{\varepsilon }\,}^!{\mathbb{C}}_{{\mathfrak{M}}({\mathbf{w}})^A} \overset{.}{=}{\mathbb{C}}_{\widetilde{S}}$ since $\widetilde{S}$ is a nonsingular submanifold of ${\mathfrak{M}}({\mathbf{w}})^A$. Applying the Verdier duality, we have

$$\begin{equation*} \operatorname {Hom}\left(H^*(i^{S!}_x \varepsilon ^! \pi ^A_*{\mathbb{C}}_{{\mathfrak{M}}({\mathbf{w}})^A}), {\mathbb{C}}\right) \overset{.}{=}H^*((i^{S!}_x \pi ^S_* {\mathbb{C}}_{\widetilde{S}})^\vee ) \overset{.}{=}H^*(i^{S*}_x \pi ^S_* {\mathbb{C}}_{\widetilde{S}}). \end{equation*}$$

Hence 14.3.7 becomes

$$\begin{equation} \operatorname {Im}\left[ M_{x,a} \to M_{x,a}^* \right] \overset{.}{=}L_{(\rho _x,{\mathbb{C}}_{\rho _x})}, {\tag{14.3.8}} \end{equation}$$

where $M_{x,a}^*$ is the dual space of $M_{x,a}$ as a complex vector space. Let us introduce an $\mathcal{A}$-module on $M_{x,a}^*$ by

$$\begin{equation*} \langle a \ast h, \xi \rangle = \langle h, (\omega _* a)\ast \xi \rangle , \quad a\in \mathcal{A}, \ h\in M_{x,a}^*, \ \xi \in M_{x,a}, \end{equation*}$$

where $\langle \ ,\ \rangle$ denotes the dual pairing, $\omega \colon Z({\mathbf{w}})^A\to Z({\mathbf{w}})^A$ is the exchange of two factors of $Z({\mathbf{w}})^A = {\mathfrak{M}}({\mathbf{w}})^A\times _{{\mathfrak{M}}_0(\infty ,{\mathbf{w}})^A}{\mathfrak{M}}({\mathbf{w}})^A$, and $\omega _*$ is the induced homomorphism on $\mathcal{A} = H_*(Z({\mathbf{w}})^A,{\mathbb{C}})$. Then 14.3.8 is compatible with $\mathcal{A}$-module structures (cf. 13, paragraphs preceding 8.6.25).

The decomposition 13.4.4 induces a similar one for $M_{x,a}^*$:

$$\begin{equation*} M_{x,a}^* = \bigoplus _\rho \operatorname {Hom}(H_*({\mathfrak{M}}(\rho )_x,{\mathbb{C}}),{\mathbb{C}}). \end{equation*}$$

The homomorphism $M_{x,a}\to M_{x,a}^*$ respects the decomposition, and induces a decomposition on 14.3.8.

Recall that we have the distinguished vector $[x]$ in $M_{x,a}$. The component $H_*({\mathfrak{M}}(\rho _x),{\mathbb{C}})$ of $M_{x,a}$ is $1$-dimensional space ${\mathbb{C}}[x]$. (See §13.3.) By the above discussion, $[x]$ is not annihilated by the above homomorphism $M_{x,a} \to M_{x,a}^*$. Thus we may consider $[x]$ also as an element of $M_{x,a}^*$.

We want to show that any nonzero ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})$-submodule $L'$ of $L_{(\rho _x,{\mathbb{C}}_{\rho _x})}$ is $L_{(\rho _x,{\mathbb{C}}_{\rho _x})}$ itself. Our strategy is the same as in the proof of Theorem 14.1.2. Since we already show that $L_{(\rho _x,{\mathbb{C}}_{\rho _x})}$ is a quotient of $M_{x,a}$, Proposition 13.3.1(3) implies $L_{(\rho _x,{\mathbb{C}}_{\rho _x})} = {\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})^-\ast [x]$. Thus it is enough to show that $L'$ contains $[x]$. To show this, consider

$$\begin{equation*} M_{x,a}^{*\circ } \overset{\operatorname {\scriptstyle def.}}{=}\{ m^*\in M^*_{x,a} \mid \text{$e_{k,r}\ast m^* = 0$ for any $k\in I$, $r\in {\mathbb{Z}}$} \}. \end{equation*}$$

By the argument as in the proof of Theorem 14.1.2, $L'$ contains a nonzero vector in $M_{x,a}^{*\circ }$. Hence it is enough to show that $M_{x,a}^{*\circ } = {\mathbb{C}}[x]$.

As in the proof of Theorem 14.1.2 above, $M_{x,a}^{*\circ }$ is a direct sum of generalized eigenspaces for $\Delta _*{\textstyle \bigwedge }_u V_l$. Let us choose and fix a direct summand $M_{x,a}^{*\circ \circ }$ contained in $H_*({\mathfrak{M}}(\rho )_x,{\mathbb{C}})$. Then $m^*\in M_{x,a}^{*\circ \circ }$ satisfies

$$\begin{equation*} \langle f_{k,r}\ast m, m^*\rangle = 0 \end{equation*}$$

for any $k$, $r$. Since $M_{x,a} = {\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})^-\ast [x]$ by Proposition 13.3.1(3), the above equation implies that $m^*\in \operatorname {Hom}({\mathbb{C}}[x],{\mathbb{C}})$. Thus we get $M_{x,a}^{*\circ } = {\mathbb{C}}[x]$ as desired.

We have shown the statement (4) during the above discussion.

Let us prove (5). Since $\pi ^A$ is a locally trivial topological fibration on each stratum ${\mathfrak{M}_0^{\operatorname {reg}}}(\rho )$, $M_{x,a}$ and $M_{y,a}$ are isomorphic if both $x$ and $y$ are contained in ${\mathfrak{M}_0^{\operatorname {reg}}}(\rho )$. Conversely, if $M_{x,a}$ and $M_{y,a}$ are isomorphic as ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})$-modules, the corresponding l-highest weights $\chi _a\left({\textstyle \bigwedge }_{-u} C_{k,x}^\bullet \right)$ and $\chi _a\left({\textstyle \bigwedge }_{-u} C_{k,y}^\bullet \right)$ are equal. Since $\chi _a\left({\textstyle \bigwedge }_{-u} C_{k,x}^\bullet \right)$ determines the homomorphism $\rho$ as in the proof of (2), $x$ and $y$ are in the same stratum.

■

Remark 14.3.9.

The assumption that $\varepsilon$ is not a root of unity is used to apply Theorem 5.5.6 and to have the invertibility of the $\varepsilon$-analogue of the Cartan matrix. It seems likely that Theorem 5.5.6 holds even if $\varepsilon$ is a root of unity. The latter condition was used to parametrize the index set of $\rho$ (i.e., Theorem 14.3.2(2)). But one should have a similar parametrization if one replaces a notion of l-weights in a suitable way. Thus Theorem 14.3.2 should hold even if $\varepsilon$ is a root of unity, if one replaces the statement (2).

Let $\mathcal{P} = \{ P(u) = (P_k(u))_k \}$ be the set of l-weights of $M_{0,a}$, which are l-dominant. Since the index set $\{ \rho \}$ of the stratum coincides with $\mathcal{P}$, we may write $L_{(\rho ,{\mathbb{C}}_\rho )}$ as $L(P)$, when ${\mathfrak{M}_0^{\operatorname {reg}}}(\rho )$ corresponds to $P\in \mathcal{P}$. The standard module $M_{x,a}$ depends only on the stratum containing $x$, so we may also write $M_{x,a}$ as $M(P)$. We have an analogue of the Kazhdan-Lusztig multiplicity formula:

Theorem 14.3.10.

Assume $\varepsilon$ is not a root of unity.

For $x\in {\mathfrak{M}}_0(\infty ,{\mathbf{w}})^A$, let $i_x\colon \{x\} \to {\mathfrak{M}}_0(\infty ,{\mathbf{w}})^A$ denote the inclusion. Let $P\in \mathcal{P}$ be the l-weight corresponding to the stratum ${\mathfrak{M}_0^{\operatorname {reg}}}(\rho _x)$ containing $x$. In the Grothendieck group of finite dimensional ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})$-modules, we have

$$\begin{equation*} M(P) = \bigoplus _{Q\in \mathcal{P}} L(Q) \otimes H^*(i_x^! IC({\mathfrak{M}_0^{\operatorname {reg}}}(\rho _Q))), \end{equation*}$$

where ${\mathfrak{M}_0^{\operatorname {reg}}}(\rho _Q)$ is the stratum corresponding to $Q\in \mathcal{P}$, and $IC({\mathfrak{M}_0^{\operatorname {reg}}}(\rho _Q))$ is the intersection cohomology complex attached to ${\mathfrak{M}_0^{\operatorname {reg}}}(\rho _Q)$ and the constant local system ${\mathbb{C}}_{{\mathfrak{M}_0^{\operatorname {reg}}}(\rho _Q)}$. Here the ${\mathbf{U}}_{\varepsilon }({\mathbf{L}}{\mathfrak{g}})$-module structure on the right hand side is given by $a\colon \xi \otimes \xi '\mapsto a\xi \otimes \xi '$.

This follows from Theorem 14.3.2 and a result in the previous subsection.

Remark 14.3.11.

By 13, 8.7.8 and Theorem 7.4.1 with Theorem 3.3.2, the cohomology group $H^{d_Q + n}(i_x^! IC({\mathfrak{M}_0^{\operatorname {reg}}}(\rho _Q)))$ vanishes for all odd $n$, where $d_Q$ is the dimension of ${\mathfrak{M}_0^{\operatorname {reg}}}(\rho _Q)$.

15. The ${\mathbf{U}}_{\varepsilon }({\mathfrak{g}})$-module structure

In this section, we assume the graph is of type $ADE$. The result of this section holds even if $\varepsilon$ is a root of unity, if we replace the simple module $L(\Lambda )$ by the corresponding Weyl module (see 10, 11.2 for the definition).

15.1

For a given ${\mathbf{w}}\in \bigoplus {\mathbb{Z}}_{\ge 0}\Lambda _k$, let $\mathcal{V}_{{\mathbf{v}}^0}({\mathbf{w}})$ be the finite set consisting of all ${\mathbf{v}}\in \bigoplus {\mathbb{Z}}_{\ge 0}\alpha _k$ such that ${\mathbf{w}}- {\mathbf{v}}$ is dominant and the weight space with weight ${\mathbf{w}}- {\mathbf{v}}^0$ is nonzero in the simple highest weight ${\mathbf{U}}_q({\mathfrak{g})}$-module $L({\mathbf{w}}-{\mathbf{v}})$.

Let $\mathcal{V}({\mathbf{w}})$ be the union of all $\mathcal{V}_{{\mathbf{v}}^0}({\mathbf{w}})$ for various ${\mathbf{v}}^0$. It is the set consisting of all ${\mathbf{v}}$ such that ${\mathbf{w}}- {\mathbf{v}}$ is dominant.

Since the graph is of type $ADE$, we have ${\mathfrak{M}}_0(\infty ,{\mathbf{w}}) = \bigsqcup _{{\mathbf{v}}} {\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}})$. Since ${\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}})$ is isomorphic to an open subvariety of ${\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})$, ${\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}})$ is irreducible if ${\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})$ is connected. Although we do not know whether ${\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})$ is connected or not (see §7.5), we consider the intersection cohomology complex $IC({\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}}))$ attached to ${\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}})$ and the constant local system ${\mathbb{C}}_{{\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}})}$. It may not be a simple perverse sheaf if ${\mathfrak{M}}({\mathbf{v}},{\mathbf{w}})$ is not connected.

We prove the following in this section:

Theorem 15.1.1.

As a ${\mathbf{U}}_{\varepsilon }({\mathfrak{g}})$-module, we have the following decomposition:

$$\begin{equation*} \operatorname {Res}M_{x,a} = \bigoplus _{{\mathbf{v}}\in \mathcal{V}({\mathbf{w}})} H^*(i_x^! IC({\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}})))\otimes L({\mathbf{w}}- {\mathbf{v}}), \end{equation*}$$

where $i_x\colon \{x\}\to {\mathfrak{M}}_0(\infty ,{\mathbf{w}})$ is the inclusion, and ${\mathbf{U}}_{\varepsilon }({\mathfrak{g}})$ acts trivially on the factor $H^*(i_x^! IC({\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}}^0,{\mathbf{w}})))$.

Remark 15.1.2.

By 13, 8.7.8 and Theorem 7.3.5 with Theorem 3.3.2, the cohomology group $H^{\operatorname {odd}}(i_x^! IC({\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}})))$ vanishes.

15.2. Reduction to $\varepsilon =1$

Suppose that $x$ is contained in a stratum ${\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}})$. Take a representative $(B,i,j)$ of $x$ and define $\rho (a)$ as in 4.1.1. Here $a$ is fixed and we do not consider $A$. We choose $S\in {\mathfrak{g}}_{\mathbf{w}}= \operatorname {Lie} G_{\mathbf{w}}$, $R\in {\mathfrak{g}}_{\mathbf{v}}= \operatorname {Lie} G_{\mathbf{v}}$, $E\in {\mathbb{C}}$ so that $\exp S = s$, $\exp R = \rho (a)$, $\exp E = \varepsilon$, where $a = (s,\varepsilon )$. Let $a_t = (\exp tS, \exp tE)$ for $t\in {\mathbb{C}}$. Then we have

$$\begin{equation*} a_t \ast (B,i,j) = \exp (tR)^{-1}\cdot (B,i,j) \end{equation*}$$

from 4.1.1. If $A_x$ denotes the stabilizer of $x$ in $G_{\mathbf{w}}\times {\mathbb{C}}^*$, the above equation means that $a_t\in A_x$.

Let us consider a ${\mathbf{U}}_{\exp tE}({\mathbf{L}}{\mathfrak{g}})$-module

$$\begin{equation*} M_t \overset{\operatorname {\scriptstyle def.}}{=}K^{A_x}({\mathfrak{M}}({\mathbf{w}})_x)\otimes _{R(A_x)} {\mathbb{C}}_{a_t} \end{equation*}$$

parametrized by $t\in {\mathbb{C}}$, where ${\mathbb{C}}_{a_t}$ is an $R(A_x)$-module given by the evaluation at $a_t$ as in §13. When $t = 1$, we can replace $A_x$ by $A$ by Theorem 7.3.5 and Theorem 3.3.2, hence the module $M_{t=1}$ coincides with $M_{x,a}$. Moreover, it depends continuously on $t$, also by Theorem 7.3.5.

Let us consider $M_t$ as a ${\mathbf{U}}_{\exp tE}({\mathfrak{g}})$-module by the restriction. Since finite dimensional ${\mathbf{U}}_{\exp tE}({\mathfrak{g}})$-modules are classified by discrete data (highest weights), it is independent of $t$. (Simple modules $L(\Lambda )$ of ${\mathbf{U}}_{\exp tE}({\mathfrak{g}})$ depend continuously on $t$.) Thus it is enough to decompose $M_t$ when $t = 0$, i.e., $s = 1$, $\varepsilon = 1$. By Theorem 7.3.5 and Theorem 3.3.2, $K^{A_x}({\mathfrak{M}}({\mathbf{w}})_x)$ is specialized to $H_*({\mathfrak{M}}({\mathbf{w}})_x,{\mathbb{C}})$ at $s=1$, $\varepsilon =1$. Thus our task now becomes the decomposition of $H_*({\mathfrak{M}}({\mathbf{w}})_x,{\mathbb{C}})$ into simple ${\mathfrak{g}}$-modules.

15.3

When $Y$ is pure dimensional, we denote by $H_{\operatorname {top}}(Y,{\mathbb{C}})$ the top degree part of $H_*(Y,{\mathbb{C}})$, that is, the subspace spanned by the fundamental classes of irreducible components of $Y$. Suppose that $Y$ has several connected components $Y_1$, $Y_2$, … such that each $Y_i$ is pure dimensional, but $\dim Y_i$ may change for different $i$. Then we define $H_{\operatorname {top}}(Y,{\mathbb{C}})$ as $\bigoplus H_{\operatorname {top}}(Y_i,{\mathbb{C}})$. Note that the degree $\operatorname {top}$ may differ for different $i$ since the dimensions are changing.

By 45, 9.4, there is a homomorphism

$$\begin{equation*} {\mathbf{U}}_{\varepsilon = 1}({\mathfrak{g}})\to H_{\operatorname {top}}(Z({\mathbf{w}}),{\mathbb{C}}). \end{equation*}$$

In fact, it is the restriction of the homomorphism in 13.4.2 for $A = \{1\}$, $\varepsilon = 1$, composed with the projection

$$\begin{equation*} H_*(Z({\mathbf{w}}),{\mathbb{C}}) \to H_{\operatorname {top}}(Z({\mathbf{w}}),{\mathbb{C}}). \end{equation*}$$

For each ${\mathbf{v}}$, we take a point $x_{\mathbf{v}}\in {\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}})$. (By Lemma 2.9.4(2), ${\mathbf{w}}- {\mathbf{v}}$ is dominant if ${\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}})$ is nonempty.) By 45, 10.2, $H_{\operatorname {top}}({\mathfrak{M}}({\mathbf{w}})_{x_{{\mathbf{v}}}},{\mathbb{C}})$ is the simple highest weight module $L({\mathbf{w}}-{\mathbf{v}})$ via this homomorphism. (In fact, we have already proved a similar result, i.e., Proposition 13.3.1.)

Proposition 15.3.1.

Consider the map $\pi \colon {\mathfrak{M}}({\mathbf{v}}^0,{\mathbf{w}})\to {\mathfrak{M}}_0({\mathbf{v}}^0,{\mathbf{w}})$. Then $\pi$, as a map into $\pi ({\mathfrak{M}}({\mathbf{v}},{\mathbf{w}}))$, is semi-small and all strata are relevant, namely

$$\begin{equation*} 2 \dim {\mathfrak{M}}({\mathbf{v}}^0,{\mathbf{w}})_{x_{{\mathbf{v}}}} = \operatorname {codim}{\mathfrak{M}}_0({\mathbf{v}},{\mathbf{w}}) \quad \text{for $x_{{\mathbf{v}}}\in {\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}})$}, \end{equation*}$$

where $\operatorname {codim}$ is the codimension in $\pi ({\mathfrak{M}}({\mathbf{v}},{\mathbf{w}}))$.

Proof.

See 44, 6.11 and 45, 10.11.

■

Proposition 15.3.2.

We have

$$\begin{equation} \pi _*\left({\mathbb{C}}_{{\mathfrak{M}}({\mathbf{v}}^0,{\mathbf{w}})}[\dim {\mathfrak{M}}({\mathbf{v}}^0,{\mathbf{w}})]\right) = \bigoplus _{{\mathbf{v}}\in \mathcal{V}_{{\mathbf{v}}^0}({\mathbf{w}})} H_{\operatorname {top}}({\mathfrak{M}}({\mathbf{v}}^0,{\mathbf{w}})_{x_{{\mathbf{v}}}},{\mathbb{C}})\otimes IC({\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}})), {\tag{15.3.3}} \end{equation}$$

where $x_{{\mathbf{v}}}$ is taken from ${\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}})$. (By Theorem 3.3.2 $H_{\operatorname {top}}({\mathfrak{M}}({\mathbf{v}}^0,{\mathbf{w}})_{x_{{\mathbf{v}}}},{\mathbb{C}})$ is independent of the choice of $x_{{\mathbf{v}}}$.)

Proof.

By the decomposition theorem for a semi-small map 13, 8.9.3, the left hand side of 15.3.3 decomposes as

$$\begin{equation*} \pi _*\left({\mathbb{C}}_{{\mathfrak{M}}({\mathbf{v}}^0,{\mathbf{w}})}[\dim {\mathfrak{M}}({\mathbf{v}}^0,{\mathbf{w}})]\right) = \bigoplus _{{\mathbf{v}},\alpha ,\phi } L_{({\mathbf{v}},\alpha ,\phi )} \otimes IC({\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}})^{\alpha },\phi ), \end{equation*}$$

where ${\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}})^{\alpha }$ is a component of ${\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}})$ and $IC({\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}})^{\alpha },\phi )$ is the intersection complex associated with an irreducible local system $\phi$ on ${\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}})^{\alpha }$. Moreover, by 13, 8.9.9, we have

$$\begin{equation} H_{\operatorname {top}}({\mathfrak{M}}({\mathbf{v}}^0,{\mathbf{w}})_{x_{{\mathbf{v}}}},{\mathbb{C}}) = \bigoplus _\phi L_{({\mathbf{v}},\alpha ,\phi )}, {\tag{15.3.4}} \end{equation}$$

where $\phi$ runs over the set of irreducible local systems on the component of ${\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}})^{\alpha }$ containing $x_{{\mathbf{v}}}$. But as argued in the proof of Theorem 14.3.2, the indecomposability of $H_{\operatorname {top}}({\mathfrak{M}}({\mathbf{w}})_{x_{{\mathbf{v}}}},{\mathbb{C}})$ implies that no intersection complex associated with a nontrivial local system appears in the summand. Moreover the left hand side of 15.3.4 is independent of the choice of the component by Theorem 3.3.2. Thus we can combine the summation over $\alpha$ together as

$$\begin{equation*} \bigoplus _{\alpha ,\phi } L_{{\mathbf{v}},\alpha ,\phi } \otimes IC({\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}})^{\alpha },\phi ) = H_{\operatorname {top}}({\mathfrak{M}}({\mathbf{v}}^0,{\mathbf{w}})_{x_{{\mathbf{v}}}},{\mathbb{C}})\otimes IC({\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}})). \end{equation*}$$

Our remaining task is to identify the index set of ${\mathbf{v}}$. The fundamental class $[{\mathfrak{M}}({\mathbf{v}}^0,{\mathbf{w}})_{x_{{\mathbf{v}}}}]$ is nonzero if ${\mathfrak{M}}({\mathbf{v}}^0,{\mathbf{w}})_{x_{{\mathbf{v}}}}$ is nonempty. Thus ${\mathfrak{M}}({\mathbf{v}}^0,{\mathbf{w}})_{x_{{\mathbf{v}}}}$ is nonempty if and only if

$$\begin{equation*} H_{\operatorname {top}}({\mathfrak{M}}({\mathbf{v}}^0,{\mathbf{w}})_{x_{{\mathbf{v}}}},{\mathbb{C}})\neq 0. \end{equation*}$$

By 45, 10.2 and the construction, $H_{\operatorname {top}}({\mathfrak{M}}({\mathbf{v}}^0,{\mathbf{w}})_{x_{{\mathbf{v}}}},{\mathbb{C}})$ is isomorphic to the weight space of weight ${\mathbf{w}}- {\mathbf{v}}^0$ in $L({\mathbf{w}}-{\mathbf{v}})$. Thus it is nonzero if and only if ${\mathbf{v}}\in \mathcal{V}_{{\mathbf{v}}^0}({\mathbf{w}})$.

■

Take $x\in {\mathfrak{M}}(\infty ,{\mathbf{w}})$ and consider the inclusion $i_x\colon \{x\} \to {\mathfrak{M}}_0(\infty ,{\mathbf{w}})$. Applying $H^*(i_x^!\bullet )$ to 15.3.3 and then summing with respect to ${\mathbf{v}}^0$, we get

$$\begin{equation} H_*({\mathfrak{M}}({\mathbf{w}})_x,{\mathbb{C}}) = \bigoplus _{{\mathbf{v}}\in \mathcal{V}({\mathbf{w}})} H_{\operatorname {top}}({\mathfrak{M}}({\mathbf{w}})_{x_{{\mathbf{v}}}},{\mathbb{C}})\otimes H^*(i_x^!IC({\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}}))). {\tag{15.3.5}} \end{equation}$$

By the convolution product, $H_{\operatorname {top}}({\mathfrak{M}}({\mathbf{w}})_{x_{{\mathbf{v}}}},{\mathbb{C}})$ is a module of $H_{\operatorname {top}}(Z({\mathbf{w}}),{\mathbb{C}})$. By 13, §8.9, the decomposition 15.3.5 is compatible with the module structure, where $H_{\operatorname {top}}(Z({\mathbf{w}}),{\mathbb{C}})$ acts on $H_{\operatorname {top}}({\mathfrak{M}}({\mathbf{w}})_{x_{{\mathbf{v}}}},{\mathbb{C}})\otimes H^*(i_x^!IC({\mathfrak{M}_0^{\operatorname {reg}}}({\mathbf{v}},{\mathbf{w}})))$ by $z\colon \xi \otimes \xi ' \mapsto z\xi \otimes \xi '$. This completes the proof of Theorem 15.1.1.

Added in proof

Crawley-Boevey recently proved that $\mathfrak{M}({\mathbf{v}},{\mathbf{w}})$ is connected (see §7.5) in “Geometry of the moment map for representations of quivers”, to appear in Compositio Math.