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.

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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

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Proposition 2.3.8.

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

Proof.

See 44, 5.5, 5.8.

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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.

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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).

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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.

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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.)

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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.

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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