On categories $\mathcal{O}$ of quiver varieties overlying the bouquet graphs

Abstract

We study representation theory of quantizations of Nakajima quiver varieties associated to bouquet quivers. We show that there are no finite dimensional representations of the quantizations $\overline{\mathcal{A}}_{\lambda }(n, \ell )$ if both $\operatorname {dim}V=n$ and the number of loops $\ell$ are greater than $1$. We show that when $n\leq 3$ there is a Hamiltonian torus action with finitely many fixed points, provide the dimensions of Hom-spaces between standard objects in category $\mathcal{O}$ and compute the multiplicities of simples in standards for $n=2$ in case of one-dimensional framing and generic one-parameter subgroups. We establish the abelian localization theorem and find the values of parameters, for which the quantizations have infinite homological dimension.

1. Introduction

Our primary goal is to study category $\mathcal{O}$ of quantizations of the Nakajima quiver variety with underlying quiver $Q=B_{\ell }$, which has one vertex, $\ell$ loops, where $\ell \in \mathbb{Z}_{\geq 0}$ and a one-dimensional framing. The notion of category $\mathcal{O}$ in the context of conical symplectic resolutions was introduced in 8. In particular in 20 the author studies the properties of category $\mathcal{O}$ for the Gieseker varieties. These are the framed moduli spaces of torsion free sheaves on $\mathbb{P}^2$ with rank $r$ and second Chern class $n$. They admit a description as quiver varieties for the quiver with one vertex, one loop, $n$-dimensional space assigned to the vertex and an $r$-dimensional framing (see Chapter $2$ of 27 for details). The results and methods of 20 provide invaluable tools for our research. We start by recalling the setup.

1.1. Generalities on category $\mathcal{O}$ for conical symplectic resolutions

We fix the base field to be $\mathbb{C}$. Recall that an affine variety $Y$ is Poisson provided it comes equipped with an algebraic Poisson bracket, i.e. a bilinear map

$$\begin{equation*} \{\cdot ,\cdot \}: \Lambda ^2\mathbb{C}[Y]\rightarrow \mathbb{C}[Y], \end{equation*}$$

s.t. for any $f,g,h \in \mathbb{C}[Y]$

•

$\{f,\{g,h\}\}+\{h,\{f,g\}\}+\{g,\{h,f\}\}=0$, the Jacobi identity;

•

$\{fg,h\}=g\{f,h\}+h\{g,f\}$, the Leibnitz rule.

Let $X_0$ be a normal Poisson affine variety equipped with an action of the multiplicative group $\mathbb{S}\coloneq \mathbb{C}^{*}$, s.t. the Poisson bracket has a negative degree with respect to this action, i.e.

$$\begin{equation*} \{\mathbb{C}[X_0]_i,\mathbb{C}[X_0]_j\}\subseteq \mathbb{C}[X_0]_{i+j-d} \text{ with } d \in \mathbb{Z}_{>0}. \end{equation*}$$

We assume that $\mathbb{C}[X_0]=\underset{i \geq 0}{\bigoplus }\mathbb{C}[X_0]_{i}$ with $\mathbb{C}[X_0]_{0}=\mathbb{C}$ w.r.t. the grading coming from the $\mathbb{S}$-action (this action will be called the contracting action). Geometrically this means that there is a unique fixed point $o\in X_0$ and the entire variety is contracted to this point by the $\mathbb{S}$-action. Let $(X,\omega )$ be a symplectic variety and $\rho :X\rightarrow X_0$ a projective resolution of singularities, which is also a morphism of Poisson varieties. In addition, assume that the action of $\mathbb{S}$ admits a $\rho$-equivariant lift to $X$. A pair $(X,\rho )$ as above is called a conical symplectic resolution.

Definition 1.1.

Let $(X,\rho )$ be a conical symplectic resolution. A quantization of the affine variety $X_0$ is a filtered algebra $\mathcal{A}$ together with an isomorphism $gr\mathcal{A}\xrightarrow {\sim }\mathbb{C}[X_0]$ of graded Poisson algebras (Poisson bracket $gr\mathcal{A}$ is given in Remark 1.3). By a quantization of $X$ we understand a sheaf (in the conical topology, i.e. open spaces are Zariski open and $\mathbb{S}$-stable) of filtered algebras $\widetilde{\mathcal{A}}$ (the filtration is complete and separated) together with an isomorphism $gr\widetilde{\mathcal{A}}\xrightarrow {\sim }\mathcal{O}_{X}$ of sheaves of graded Poisson algebras.

Remark 1.2.

There are sufficiently many $\mathbb{S}$-stable open affine subsets. Namely, due to a result of Sumihiro every point $x$ of $X$ has an open affine neighborhood in the conical topology (see Section $3$, Corollary $2$ in 31).

Remark 1.3.

We would like to point out that the algebra $A\coloneq gr\mathcal{A}$ has a natural Poisson bracket. Let $a \in A_i$ and $b \in A_j$ with $\tilde{a}\in \mathcal{A}_{\leq i}$ and $\tilde{b}\in \mathcal{A}_{\leq j}$ any lifts, then the Poisson bracket is given by

$$\begin{equation*} \{a,b\}\coloneq [\tilde{a},\tilde{b}]+\mathcal{A}_{i+j-d-1}, \end{equation*}$$

where $d\in \mathbb{Z}_{\geq 0}$ is the maximal positive integer, s.t. $[a',b']\in \mathcal{A}_{i+j-d-1}$ for any $a' \in \mathcal{A}_{i}, b' \in \mathcal{A}_{j}$, called the degree (notice that $[\tilde{a},\tilde{b}]\in \mathcal{A}_{i+j-1}$ since the algebra $A$ is isomorphic to $\mathbb{C}[X_0]$ and hence commutative). It is this bracket that we want to match the original bracket on $\mathbb{C}[X_0]$ in Definition 1.1.

Remark 1.4.

There is a map from the set of quantizations of $X$ to the second de Rham cohomology $H^2_{DR}(X)$. This map is called the period map and is an isomorphism provided $H^i(X,\mathcal{O}_X)=0$ for all $i>0$ (see 4). If this is the case, the quantizations $\widetilde{\mathcal{A}}$ are parameterized (up to isomorphism) by the points of $H^2_{DR}(X)$. The quantization corresponding to the cohomology class $\lambda$ will be denoted by $\widetilde{\mathcal{A}}_{\lambda }$.

Suppose that $X$ is equipped with a Hamiltonian action of a torus $T$ with finitely many fixed points, i.e. $|X^T| < \infty$. Assume, in addition, that the action of $T$ commutes with the contracting action of $\mathbb{S}$. A one-parametric subgroup $\nu : \mathbb{C}^{*}\rightarrow T$ is called generic if $X^T=X^{\nu (\mathbb{C}^{*})}$. To a generic one-parametric subgroup $\nu : \mathbb{C}^{*}\rightarrow T$ one can associate a category of modules over the algebra $\mathcal{A}$ defined above, called category $\mathcal{O}_{\nu }(\mathcal{A})$. Namely, the action of $\nu$ lifts to $\mathcal{A}$ and induces a grading on it, i.e. $\mathcal{A}=\underset{i \in \mathbb{Z}}{\bigoplus }\mathcal{A}_{i,\nu }$. We denote

$$\begin{align} & \mathcal{A}^{\geq 0,\nu }=\underset{i \geq 0}{\bigoplus }\mathcal{A}_{i,\nu }, \mathcal{A}^{\leq 0,\nu }=\underset{i \leq 0}{\bigoplus }\mathcal{A}_{i,\nu }\nobreakspace (\text{similarly define } \mathcal{A}^{< 0,\nu },\mathcal{A}^{> 0,\nu }) \text{ and } \tag{1}\\ & \mathcal{C}_{\nu }(\mathcal{A})\coloneq \mathcal{A}^{\geq 0,\nu }/\left(\mathcal{A}^{\geq 0,\nu }\cap \mathcal{A}\mathcal{A}^{>0,\nu }\right)=\mathcal{A}_{0}/\underset{i > 0}{\bigoplus }\mathcal{A}_{-i}\mathcal{A}_{i}. \tag{2} \end{align}$$

Let $\mathcal{A}\operatorname {-mod}$ be the category of finitely generated $\mathcal{A}$-modules.

Definition 1.5.

The category $\mathcal{O}_{\nu }(\mathcal{A})$ is the full subcategory of $\mathcal{A}\operatorname {-mod}$, on which $\mathcal{A}^{\geq 0,\nu }$ acts locally finitely.

Recall that if $\mathcal{R}$ is a commutative Noetherian ring and $X=Spec \mathcal{R}$, then one has an equivalence of abelian categories:

$$\begin{equation} \mathcal{R}\operatorname {-mod} \mathrel{\mathop{\rightleftarrows }^{\mathrm{Loc}}_{\mathrm{\Gamma }}} Coh(X), {\tag{3}} \end{equation}$$

where $\Gamma$ and $Loc$ are the functor of global sections and localization respectively (see Chapter $II$, Corollary $5.5$ in 16 for details).

Definition 1.6.

An $\widetilde{\mathcal{A}}_\lambda$-module $M$ is called coherent provided there is a global complete and separated filtration on $M$, s.t. $gr (M)$ is a coherent $\mathcal{O}_{X}$-module. The category of coherent $\widetilde{\mathcal{A}}_\lambda$-modules will be denoted by $Coh(\widetilde{\mathcal{A}}_\lambda )$ (or simply $\widetilde{\mathcal{A}_{\lambda }}\operatorname {-mod}$).

The noncommutative analogue of equivalence 3 is

$$\begin{equation} \mathcal{A}_\lambda \operatorname {-mod} \mathrel{\mathop{\rightleftarrows }^{\mathrm{Loc_\lambda }}_{\mathrm{\Gamma _\lambda }}} Coh(\widetilde{\mathcal{A}_\lambda }), {\tag{4}} \end{equation}$$

here $\mathcal{A}_\lambda \coloneq \Gamma (X,\widetilde{\mathcal{A}_\lambda })$ (notice that $\mathcal{A}_\lambda$ is a quantization of $X_0$ provided $\widetilde{\mathcal{A}_\lambda }$ is a quantization of $X$). The equivalence 4 has a weaker (derived form):

$$\begin{equation} D^b(\mathcal{A}_\lambda \operatorname {-mod}) \mathrel{\mathop{\rightleftarrows }^{\mathrm{LLoc_\lambda }}_{\mathrm{R\Gamma _\lambda }}} D^b(Coh(\widetilde{\mathcal{A}}_\lambda )). {\tag{5}} \end{equation}$$

Definition 1.7.

If the functors $\Gamma _\lambda$ and $Loc_\lambda$ are mutually inverse equivalences, we say that abelian localization holds for $\lambda$ and if $R\Gamma _\lambda$ and $LLoc_\lambda$ are quasi-inverse equivalences (between the bounded derived categories) that derived localization holds.

Example 1.8.

Let $\mathfrak{g}$ be a simple Lie algebra with Borel subalgebra $\mathfrak{b}$ and Cartan subalgebra $\mathfrak{h}$. In order to fit the classical BGG category $\mathcal{O}$ in this framework, one needs to consider the Springer resolution $X=T^*(G/B)\rightarrow \mathcal{N}=X_0$ of the nilpotent cone $\mathcal{N}\subset \mathfrak{g}^*$. Recall that an element $x \in \mathfrak{g}$ is called nilpotent if the operator $ad^*_x: \mathfrak{g}^* \rightarrow \mathfrak{g}^*$ is nilpotent and $\mathcal{N}$ is the set of all nilpotent elements of $\mathfrak{g}^*$. The nilcone $\mathcal{N}$ is a Poisson variety w.r.t. the Kirillov-Kostant-Souriau bracket and the symplectic leaves in $\mathcal{N}$ are the coadjoint orbits. The tori are the maximal torus $T\subset GL(V)$ and $\mathbb{S}\coloneq \mathbb{C}^{*}$ acting by inverse scaling. Let $\mu :Z(\mathfrak{g})\rightarrow \mathbb{C}$ be a central character, then the block $\mathcal{O_{\mu }}\subset \mathcal{O}$ consists of finitely generated $U(\mathfrak{g})$-modules for which $U(\mathfrak{b})$ acts locally finitely, $U(\mathfrak{h})$ semisimply and the center with generalized character $\mu$. Pick a generic one-parameter subgroup $\nu (\mathbb{C}^{*})\subset T$, s.t. $\mathfrak{b}$ is spanned by elements with positive $\nu (\mathbb{C}^{*})$-weights. Let $U(\mathfrak{g})_{\mu }\coloneq U(\mathfrak{g})/\mathcal{I}_{\mu }$ with $\mathcal{I}_{\mu }$ the ideal generated by $z-\mu (z)$ for $z\in Z(\mathfrak{g})$ be the central reduction of $U(\mathfrak{g})$ w.r.t. the central character $\mu$.

We want to show that $U(\mathfrak{g})_{\mu }$ is a quantization of the nilcone $\mathcal{N}$. One can explicitly describe the Poisson bracket on $\mathbb{C}[\mathcal{N}]$ descending from $U(\mathfrak{g})_{\mu }$ (as explained in Remark 1.3). Recall that according to the PBW theorem $gr(U(\mathfrak{g}))$ is isomorphic to $S(\mathfrak{g})=\mathbb{C}[\mathfrak{g}^*]$. Moreover, Harish Chandra theorem asserts that $Z(\mathfrak{g})$ is isomorphic to $S(\mathfrak{h})^W=\mathbb{C}[\mathfrak{h}^*]^W$. Here $W$ is the Weyl group acting on $\mathfrak{h}^*$ via $w\cdot \mu = w(\mu +\rho )-\rho$, where $\rho$ is half the sum of all positive roots. Combining these results allows to show the isomorphism of algebras $gr(U(\mathfrak{g}))_{\mu }\simeq \mathbb{C}[\mathcal{N}]$. Let $x_1$, …, $x_n$ be a basis of $\mathfrak{g}$ and $c_{ij}^k\in \mathbb{C}$ the structure constants given by $[x_i,x_j]=\sum \limits _{k=1}^n c_{ij}^kx_k$. The Poisson bracket on $\mathcal{N}\subset \mathfrak{g}^*$ becomes the restriction of the bracket on $\mathfrak{g}^*$ given by

$$\begin{equation*} \{f,g\}=\sum _{k=1}^n c_{ij}^kx_k\frac{\partial f}{\partial x_i}\frac{\partial g}{\partial x_j}, \text{ for } f,g \in \mathbb{C}[\mathcal{\mathfrak{g}^*}] \end{equation*}$$

which can be more conveniently rewritten as

$$\begin{equation*} \{f,g\}(\xi )=\langle \xi , [d_\xi f,d_\xi g]\rangle , \end{equation*}$$

where $\xi \in \mathfrak{g}^*$, $d_\xi f\in \mathfrak{g}^{**}\simeq \mathfrak{g}$ stands for the differential of $f$ at $\xi$ and $[,]$ denotes the Lie bracket on $\mathfrak{g}$ (see Proposition $1.3.18$ in 10 for details). This is exactly the Kirillov-Kostant-Souriau bracket on the nilcone $\mathcal{N}$.

Next we want to compare the categories $\mathcal{O}_{\nu }(U(\mathfrak{g})_{\mu })$ and $\mathcal{O_{\mu }}$. The difference in the requirements for an object $M\in U(\mathfrak{g})_{\mu }\operatorname {-mod}$ to be in $\mathcal{O}_{\nu }(U(\mathfrak{g})_{\mu })$ or $\mathcal{O_{\mu }}$ is that for the former containment $Z(\mathfrak{g})$ must act on $M$ with an honest character $\mu$, while for the latter the action of $U(\mathfrak{h})$ on $M$ has to be semisimple. In case $\mu$ is dominant regular ($\langle \mu +\rho ,\alpha ^{\vee }\rangle \not \in \mathbb{Z}_{\leq 0}$ for all positive roots $\alpha$) these conditions are interchangeable, i.e. one gets an equivalent category by dropping one condition and adding the other (see Theorem $1$ in 30), and, hence, the categories $\mathcal{O}_{\nu }(U(\mathfrak{g})_{\mu })$ and $\mathcal{O_{\mu }}$ are equivalent.

Finally, let $\mathcal{D}_{\mu }(G/B)$ stand for the category of $\mu$-twisted $\mathcal{D}$-modules on the flag variety $G/B$. Then one has an equivalence

$$\begin{equation*} U(\mathfrak{g})_{\mu }\operatorname {-mod} \mathrel{\mathop{\rightleftarrows }^{\mathrm{Loc}}_{\mathrm{\Gamma }}} \mathcal{D}_{\mu }(G/B)\operatorname {-mod} \end{equation*}$$

for dominant regular $\mu$ (with $\langle \mu +\rho ,\alpha ^{\vee }\rangle \not \in \mathbb{Z}_{\leq 0}$ for all positive roots $\alpha$), this is the Beilinson-Bernstein theorem, see 1, while

$$\begin{equation*} D^b(U(\mathfrak{g})_{\mu }\operatorname {-mod}) \mathrel{\mathop{\rightleftarrows }^{\mathrm{LLoc}}_{\mathrm{R\Gamma }}} D^b(\mathcal{D}_{\mu }(G/B)\operatorname {-mod}) \end{equation*}$$

is an equivalence provided $\langle \mu +\rho ,\alpha ^{\vee }\rangle \neq 0$, see 2.

Remark 1.9.

More generally, there are two functors between the categories $\mathcal{A}_{\lambda }\operatorname {-mod}$ and $\widetilde{\mathcal{A}_{\lambda }}\operatorname {-mod}$ and the corresponding derived categories:

$$\begin{equation*} \mathcal{A}_{\lambda }\operatorname {-mod} \mathrel{\mathop{\rightleftarrows }^{\mathrm{Loc_{\lambda }}}_{\mathrm{\Gamma _{\lambda }}}} \widetilde{\mathcal{A}}_{\lambda }\operatorname {-mod}, \end{equation*}$$

$$\begin{equation*} D^{b}(\mathcal{A}_{\lambda }\operatorname {-mod}) \mathrel{\mathop{\rightleftarrows }^{\mathrm{LLoc_{\lambda }}}_{\mathrm{R\Gamma _{\lambda }}}} D^{b}(\widetilde{\mathcal{A}}_{\lambda }\operatorname {-mod}). \end{equation*}$$

Definition 1.10.

If the functors $\mathrm{Loc}_{\lambda }, \Gamma _{\lambda }$ ($\mathrm{LLoc}_{\lambda }, \mathrm{R}\Gamma _{\lambda }$) are mutually inverse equivalences, we say that abelian (derived) localization holds for the pair $(\lambda ,\widetilde{\mathcal{A}}_{\lambda })$.

1.2. Questionnaire on quantizations

Let $\rho : X\rightarrow X_0$ be a conical symplectic resolution. Assume that $X$ admits a Hamiltonian torus action with finitely many fixed points and the nonzero cohomology of the structure sheaf of $X$ vanishes. We list some typical questions that can be asked about quantizations and categories of modules thereof.

(1)

For which $\lambda$ does $\mathcal{A}_{\lambda }$ have finite homological dimension?

(2)

What is the classification of finite dimensional irreducible modules?

(3)

What are the supports of these modules?

(4)

What are the two-sided ideals of $\mathcal{A}_{\lambda }$?

(5)

For which $\lambda \in H^2_{DR}(X)$ do the abelian/derived localizations hold?

(6)

What are the composition series of standard modules in category $\mathcal{O}$?

Remark 1.11.

According to a result of McGerty and Nevins (see 23, Theorem 1.1) the ‘derived equivalence locus’ appearing in $(5)$ is the same as the locus, providing affirmative answer in $(1)$.

1.3. Main results and structure of the paper

The present paper is devoted to study of quantizations of Nakajima quiver varieties overlying the bouquet graph (one vertex and finitely many loop edges) and categories $\mathcal{O}$ thereof. Let $\overline{\mathcal{M}}^{\theta }(n,\ell )$ denote the Nakajima quiver variety for quiver $Q$ with one vertex and $\ell$ loops, a vector space $V\simeq \mathbb{C}^n$ assigned to the vertex and one-dimensional framing (see Section 2 for precise definitions and detailed explanations). The quantizations $\overline{\mathcal{A}}_{\lambda }(2, \ell )$ of $\overline{\mathcal{M}}^{\theta }(n,\ell )$ are naturally parameterized by $\lambda \in H^2(\overline{\mathcal{M}}^{\theta }(n,\ell ))\simeq \mathbb{C}$ and $\theta \in \operatorname {char}(GL(V))$.

The exposition in the paper is organized as follows. Section 2 gives preliminary results on the varieties $\overline{\mathcal{M}}^{\theta }(n,\ell )$. It is shown that $\overline{\mathcal{M}}^{\theta }(n,\ell )$ has finitely many fixed points w.r.t. the Hamiltonian torus $T$ action for $n\leq 3$ (here $T\subset GL(V)$ is a maximal torus), the central fiber of the resolution $\bar{\rho }:\overline{\mathcal{M}}^{\theta }(n,\ell )\rightarrow \overline{\mathcal{M}}(n,\ell )$ is of dimension less than $\frac{1}{2}\overline{\mathcal{M}}^{\theta }(n,\ell )$ for $n,\ell >1$. From this (using Gabber’s theorem) one deduces that there are no finite dimensional $\overline{\mathcal{A}}_{\lambda }(n,\ell )$-modules with generic $\nu$. Furthermore, the resolutions $\bar{\rho }:\overline{\mathcal{M}}^{\theta }(n,\ell )\rightarrow \overline{\mathcal{M}}(n,\ell )$ serve as counterexamples to Conjecture $1.3.1$ in 12. The explanation of this phenomenon concludes the section (see Remark 2.28 for details).

In Section 3, following the recipe of 25, 26 (see also Section $2$ of 5), the description of symplectic leaves of $\overline{\mathcal{M}}(n,\ell )$ and slices to points on them for $n=2,3$ is obtained. One of the two nontrivial slices to $\overline{\mathcal{M}}(2,\ell )$ turns out to be a hypertoric variety. The description of $T$-fixed points on that slice is provided.

Following the lines of 7, we give an overview on generalities on hypertoric varieties and categories $\mathcal{O}$ associated to them and describe category $\mathcal{O}$ for the slices (Proposition 4.15, Section 4).

The next section is devoted to the proof of Theorem 5.4 (incomplete form of abelian localization theorem) and the description of the locus of $\lambda$, for which the algebra $\overline{\mathcal{A}}_{\lambda }(2, \ell )$ has finite homological dimension (Corollary 5.5).

Then, using the construction of restriction functor introduced in 3 for rational Cherednik algebras (quantizations of the Hilbert scheme of points on $\mathbb{C}^2$) and its generalization for the Gieseker scheme in 20, we define a functor $Res:\mathcal{O}_{\nu }(\mathcal{\overline{A}}_{\lambda }(2,\ell ))\rightarrow \mathcal{O}_{\nu }(\mathcal{\overline{S}}_{\lambda }(2,\ell ))$, where $\mathcal{O}_{\nu }(\mathcal{\overline{S}}_{\lambda }(2,\ell ))$ stands for the category $\mathcal{O}$ for the slice. This functor is exact and faithful on standard objects. It serves as the main ingredient in the proof of Theorem 6.15, which gives a description of Hom-spaces between standard objects in $\mathcal{O}_{\nu }(\mathcal{\overline{A}}_{\lambda }(2,\ell ))$. The multiplicities of simples in standards are established in Corollary 6.16.

The complete form of abelian localization theorem appears in Section 7 (see Theorem 7.6).

2. First results on $\mathcal{O}_{\nu }(\mathcal{\overline{A}}_{\lambda }(n,\ell ))$

In this section we collect some basic information on the category $\mathcal{O}_{\nu }(\mathcal{\overline{A}}_{\lambda }(n,\ell ))$. The quantization of $\overline{\mathcal{M}}^{\theta }(n,\ell )$ corresponding to a character $\lambda$ of $\mathfrak{g}$ is the algebra $\mathcal{\overline{A}}_{\lambda }(n,\ell )\coloneq (D(\overline{R})/[D(\overline{R})\{\Phi (x)-\lambda (x), x \in \mathfrak{g}\}])^{G}$.

2.1. Category $\mathcal{O}$ for the quantizations of quiver varieties with $Q=B_{\ell }$

We study the Nakajima quiver variety with underlying quiver $Q$, which has one vertex, $\ell$ loops, where $\ell \in \mathbb{Z}_{\geq 0}$ and a one-dimensional framing. This variety admits the following description. One starts with a vector space $V$ of dimension $n$ and considers the space $R\coloneq \mathfrak{gl}(V)^{\oplus \ell }\oplus V^*$, which has a natural $G\coloneq GL(V)$ action. The identification of $\mathfrak{g}\coloneq \mathfrak{gl}(V)$ with $\mathfrak{g}^*$ via the trace form enables to identify the cotangent bundle $T^*R$ with $\mathfrak{gl}(V)^{\oplus 2\ell }\oplus V^* \oplus V$. Next notice that $T^*R$ is a symplectic vector space with a Hamiltonian action of $G$. The corresponding moment map is given by

$$\begin{equation} \mu (X_{1}, \mathinner{\ldotp \ldotp \ldotp }, X_{\ell },Y_{1}, \mathinner{\ldotp \ldotp \ldotp }, Y_{\ell },i,j) = \sum _{k=0}^{\ell }[X_{k},Y_{k}]-ji. {\tag{6}} \end{equation}$$

To define the Nakajima quiver variety $\mathcal{M}^{\theta }(n,\ell )$, we need to choose some character $\theta$ of $G$. It is known that $\theta$ is an integral power of the determinant, i.e. $\theta =\operatorname {det}^k$ for some $k \in \mathbb{Z}$.

Definition 2.1.

The Nakajima quiver variety $\mathcal{M}^{\theta }(n,\ell )$ is the GIT quotient $\mu ^{-1}(0)^{\theta -ss}//^{\theta }{G}$. In particular, $\mathcal{M}(n,\ell )=\mu ^{-1}(0)//G\coloneq \operatorname {Spec}\mathbb{C}[\mu ^{-1}(0)]^{G}$.

Remark 2.2.

An application of the Hilbert-Mumford criterion shows that the $\theta$-semistable locus admits the following natural description (see Lemma $3.8$ in 26 for details). Let $S\subseteq V$ be a subspace, s.t. $X_t(S), Y_t(S) \subseteq S$ for all $1 \leq t \leq \ell$, then

$$\begin{align*} & S \subset Ker\nobreakspace j\nobreakspace \Rightarrow S=0, \text{ if }\theta >0, \\ & S \supset Im\nobreakspace i\nobreakspace\nobreakspace \Rightarrow S=V, \text{ if }\theta <0. \end{align*}$$

The torus $T=(\mathbb{C}^{*})^{\ell }$ acts on $R$ by rescaling $X_{1}$, …, $X_{\ell }$. This naturally gives rise to an action on $T^*R$. This action is Hamiltonian and commutes with the action of $G$ and, therefore, descends to $\mathcal{M}(n,\ell )$ and $\mathcal{M}^{\theta }(n,\ell )$. The action of $s\in \mathbb{S}$ is given by multiplication of all the components of $x \in T^*R$ by $s^{-1}$. Similarly, it commutes with the action of $G$ and descends to $\mathcal{M}(n,\ell )$ and $\mathcal{M}^{\theta }(n,\ell )$.

For any $\theta \neq 0$ the action of $G$ on $\mu ^{-1}(0)^{\theta -ss}$ is free. This implies that the variety $\mathcal{M}^{\theta }(n,\ell )$ is smooth and symplectic and is known to be a symplectic resolution of the normal Poisson variety $\mathcal{M}(n,\ell )$. We denote by $\rho$ the corresponding map $\rho :\mathcal{M}^{\theta }(n,\ell )\rightarrow \mathcal{M}(n,\ell )$. It is a conical symplectic resolution.

Set $\overline{R}=\mathfrak{sl}(V)^{\oplus \ell }\oplus V^{*}$ and let $\overline{\mathcal{M}}(n,\ell )$ be the affine variety $\mu ^{-1}(0)//G$, where slightly abusing notation, we denote by $\mu$ the moment map for the Hamiltonian action of $G$ on $T^*\overline{R}$. Similarly, we set $\overline{\mathcal{M}}^{\theta }(n,\ell )\coloneq \mu ^{-1}(0)^{\theta -ss}//^{\theta }{G}$. Next we describe quantizations of $\overline{\mathcal{M}}(n,\ell )$. Denote the ring of differential operators on $\overline{R}$ by $D(\overline{R})$.

Definition 2.3.

A $G$-equivariant linear map $\Phi : \mathfrak{g}\rightarrow D(\overline{R})$ satisfying $[\Phi (x), a]=x_{\overline{R}}(a)$ for any $x \in \mathfrak{g}$ and $a \in D(\overline{R})$ is called a quantum comoment map.

Remark 2.4.

The quantum comoment map $\Phi$ is defined up to adding a character $\lambda : \mathfrak{g}\rightarrow \mathbb{C}$.

Notice that we can identify $D(\overline{R})$ with $D(\overline{R}^*)$ via the Fourier transform sending $\partial _r \in D(\overline{R})$ to the function $r \in D(\overline{R}^*)$ and $r^* \in D(\overline{R})$ to $-\partial _{r^*} \in D(\overline{R}^*)$. Thus defined isomorphism $D(\overline{R})\rightarrow D(\overline{R}^*)$ allows to consider two quantum comoment maps $\Phi ,\widetilde{\Phi }:\mathfrak{gl}(V)\rightarrow D(\overline{R})$ sending $x \in \mathfrak{g}$ to the corresponding vector field $x_{\overline{R}}$ or $x_{\overline{R}^*}$. Now define the symmetrized quantum comoment map to be $\Phi ^{sym}\coloneq \frac{\Phi +\widetilde{\Phi }}{2}$. A direct computation shows that $\Phi ^{sym}(x)=\Phi (x)-\zeta (x)$, where $\zeta$ is half the character of the action of $G$ on $\Lambda ^{top}R$. For our quiver $Q$ with one-dimensional framing $\zeta (x)=\frac{1}{2}tr(x)$.

Next we take a character $\lambda$ of $\mathfrak{g}$ and consider the quantizations

$$\begin{equation*} \mathcal{\overline{A}}_{\lambda }(n,\ell )\coloneq (D(\overline{R})/[D(\overline{R})\{\Phi (x)-\lambda (x), x \in \mathfrak{g}\}])^{G}, \end{equation*}$$

$$\begin{equation*} \mathcal{\overline{A}}^{sym}_{\lambda }(n,\ell )\coloneq (D(\overline{R})/[D(\overline{R})\{\Phi ^{sym}(x)-\lambda (x), x \in \mathfrak{g}\}])^{G}. \end{equation*}$$ The filtration on $\mathcal{\overline{A}}_{\lambda }(n,\ell )$ is induced from the Bernstein filtration on $D(\overline{R})$ (here deg $\overline{R}=$ deg $\overline{R}^*=1$). Recall that $\mathbb{C}[\overline{\mathcal{M}}(n,\ell )]=(\mathbb{C}[T^*\overline{R}]/I)^G$, where $I\coloneq \{\mu ^*(\xi ), \xi \in \mathfrak{g}\}$ is the ideal generated by the image of $\mathfrak{g}$ under the comoment map, and denote $\mathcal{I}_{\lambda }\coloneq \{\Phi (x)-\lambda (x), x \in \mathfrak{gl}(V)\}$. The surjectivity of the natural map $\mathbb{C}[\overline{\mathcal{M}}(n,\ell )]\rightarrow \operatorname {gr} (\mathcal{\overline{A}}_{\lambda }(n,\ell ))$ follows from the containment $I \subset \operatorname {gr} (\mathcal{I}_{\lambda })$. The reverse containment of ideals follows from the regularity of the sequence $\mu ^*(\xi _1)$, …, $\mu ^*(\xi _{n^2})$, where $\xi _1$, …, $\xi _{n^2}$ is some basis for $\mathfrak{g}$. The regularity of the sequence is equivalent to flatness of the moment map $\mu$.

We notice that the difference between $\mathcal{\overline{A}}_{\lambda }(n,\ell )$ and the algebra $\mathcal{A}_{\lambda }(n,\ell )$ (constructed analogously for $R=\mathfrak{gl}(V)^{\oplus \ell }\oplus V^{*}$) is that $\mathcal{A}_{\lambda }(n,\ell )=D(\mathbb{C}^{\ell })\otimes \mathcal{\overline{A}}_{\lambda }(n,\ell )$. Thus, some questions about representation theory of $\mathcal{\overline{A}}_{\lambda }(n,\ell )$ reduce to analogous ones for $\mathcal{A}_{\lambda }(n,\ell )$.

The quantizations $\overline{\mathcal{A}}^{\theta }$ of $\overline{\mathcal{M}}^{\theta }(n,\ell )$ are parameterized (up to isomorphism) by the points of $H^2(\overline{\mathcal{M}}^{\theta }(n,\ell ))\simeq \mathbb{C}$ (see 4). The quantization corresponding to $\lambda$ will be denoted by $\overline{\mathcal{A}}_{\lambda }^{\theta }$.

Remark 2.5.

The period of the quantization $\mathcal{\overline{A}}_{\lambda }(n,\ell )^{sym}$ is equal to $\lambda$.

We fix our choice of character $\theta =det^{-1}$. As can be inferred from the proof of Lemma 2.6, this choice is generic.

Lemma 2.6.

There is an isomorphism $\mathcal{\overline{A}}_{\lambda }(n,\ell )\cong \mathcal{\overline{A}}_{-\lambda -1}(n,\ell )$.

Proof.

There is a symplectomorphism $\gamma : \overline{\mathcal{M}}^{\theta }(n,\ell )\simeq \overline{\mathcal{M}}^{-\theta }(n,\ell )$ produced by

$$\begin{equation*} (X_{1}, \mathinner{\ldotp \ldotp \ldotp }, X_{\ell }, Y_{1}, \mathinner{\ldotp \ldotp \ldotp }, Y_{\ell },i,j) \mapsto (Y_{1}^{*}, \mathinner{\ldotp \ldotp \ldotp }, Y_{\ell }^{*}, -X_{1}^{*}, \mathinner{\ldotp \ldotp \ldotp }, -X_{\ell }^{*},j^{*},-i^{*}), \end{equation*}$$

thus inducing multiplication by $-1$ on $H^{2}(\overline{\mathcal{M}}^{\theta }(n,\ell ), \mathbb{Z})$. As the image of $\lambda$ under the period map is $\lambda +\frac{1}{2}\in H^{2}(\overline{\mathcal{M}}^{\theta }(n,\ell ), \mathbb{Z})$, the result follows.

■

Remark 2.7.

The categories $\mathcal{O}_{\nu }(\mathcal{\overline{A}}_{\lambda }(n,\ell ))$ and $\mathcal{O}_{\nu }(\mathcal{A}_{\lambda }(n,\ell ))$ are, in fact, equivalent. Indeed, recall that $\mathcal{A}_{\lambda }(n,\ell )=D(\mathbb{C}^{\ell })\otimes \mathcal{\overline{A}}_{\lambda }(n,\ell )$ and let $t_1$, …, $t_{\ell }$ be the coordinates on $\mathbb{C}^{\ell }$. Then the functor $\mathcal{O}_{\nu }(\mathcal{\overline{A}}_{\lambda }(n,\ell ))\rightarrow \mathcal{O}_{\nu }(\mathcal{A}_{\lambda }(n,\ell ))$ given by $M\mapsto \mathbb{C}[t_1,\mathinner{\ldotp \ldotp \ldotp },t_{\ell }]\otimes M$ produces an equivalence of categories. It has a quasi-inverse functor which sends $N \in \mathcal{O}_{\nu }(\mathcal{A}_{\lambda }(n,\ell ))$ to the annihilator of $\langle \partial t_1,\mathinner{\ldotp \ldotp \ldotp }, \partial t_{\ell }\rangle$.

Definition 2.8.

We have the standardization and costandardization functors $\triangle _{\nu }$ and $\nabla _{\nu }:\mathcal{C}_{\nu }(\mathcal{\overline{A}}_{\lambda }(n,\ell ))\operatorname {-mod} \rightarrow \mathcal{O}_{\nu }(\mathcal{\overline{A}}_{\lambda }(n,\ell ))$ given by

$$\begin{equation*} \triangle _{\nu }(N)\coloneq \mathcal{\overline{A}}_{\lambda }(n,\ell )/\mathcal{\overline{A}}_{\lambda }(n,\ell )\mathcal{\overline{A}}^{>0}_{\lambda }(n,\ell )\otimes _{\mathcal{C}_{\nu }(\mathcal{\overline{A}}_{\lambda }(n,\ell ))} N, \end{equation*}$$

$$\begin{equation*} \nabla _{\nu }(N)\coloneq \operatorname {Hom}_{\mathcal{C}_{\nu }(\mathcal{\overline{A}}_{\lambda }(n,\ell ))}(\mathcal{\overline{A}}_{\lambda }(n,\ell )/\mathcal{\overline{A}}^{<0}_{\lambda }(n,\ell )\mathcal{\overline{A}}_{\lambda }(n,\ell ),N). \end{equation*}$$

We consider the restricted $\operatorname {Hom}$ (w.r.t. the natural grading on

$$\begin{equation*} \mathcal{\overline{A}}_{\lambda }(n,\ell )/\mathcal{\overline{A}}^{<0}_{\lambda }(n,\ell )\mathcal{\overline{A}}_{\lambda }(n,\ell )) \end{equation*}$$

in the definition of the operator $\nabla _{\nu }$ above.

Definition 2.9.

Let $\mathcal{C}$ be an abelian, artinian category enriched over $\mathbb{R}$ with simple objects $\{S_{\alpha } | \alpha \in \mathcal{I}\}$, projective covers $\{P_{\alpha } | \alpha \in \mathcal{I}\}$, and injective hulls $\{I_{\alpha } | \alpha \in \mathcal{I}\}$. Let $\preceq$ be a partial order on the index set $\mathcal{I}$. We call $\mathcal{C}$ highest weight with respect to this partial order if there is a collection of objects $\{\triangle _{\alpha } | \alpha \in \mathcal{I}\}$ and epimorphisms $P_{\alpha }\overset{\Pi _{\alpha }}{\rightarrow } \triangle _{\alpha }\overset{\pi _{\alpha }}{\rightarrow } S_{\alpha }$ such that for each $\alpha \in \mathcal{I}$, the following conditions hold:

(1)

the object $\operatorname {ker} \pi _{\alpha }$ has a filtration such that each subquotient is isomorphic to $S_{\beta }$ for some $\beta \prec \alpha$;

(2)

the object $\operatorname {ker} \Pi _{\alpha }$ has a filtration such that each subquotient is isomorphic to $\triangle _{\gamma }$ for some $\gamma \succ \alpha$.

The objects $\triangle _{\alpha }$ are called standard objects.

Definition 2.10.

Let $\theta _1$, …, $\theta _k$ be the characters of $T$-action on the vector space $\underset{p\in X^T}{\bigoplus }T_pX$, where $T_pX$ stands for the tangent space at a fixed point $p\in X^T$. The kernels $\ker (\theta _1)$, …, $\ker (\theta _k)$ partition the cocharacter space $\operatorname {Hom}(\mathbb{C}^*,T)\otimes _\mathbb{Z}\mathbb{R}$ into polyhedral cones, to be referred to as chambers. If a cocharacter $\nu$ lies in the interior of a chamber (in other words, all $\theta _i(\nu )\neq 0$), then $X^{\nu (\mathbb{C}^*)}=X^T$. Such one-parameter subgroups are called generic.

The next result can be found in 18 (see Proposition $2.2$).

Proposition 2.11.

Suppose that abelian localization holds and $\lambda$ is generic (outside some finite set). Choose a generic one-parameter subgroup $\nu$. Then the following are true:

(1)

the category $\mathcal{O}_{\nu }(\mathcal{\overline{A}}_{\lambda }(n,\ell ))$ depends only on the chamber of $\nu$;

(2)

the natural functor $D^{b}(\mathcal{O}_{\nu }(\mathcal{\overline{A}}_{\lambda }((n,\ell )))\rightarrow D^{b}(\mathcal{\overline{A}}_{\lambda }(n,\ell )\operatorname {-mod})$ is a full embedding;

(3)

$\mathcal{C}_{\nu }(\mathcal{\overline{A}}_{\lambda }(n,\ell ))=\mathbb{C}[\overline{\mathcal{M}}^{\theta }(n,\ell )^{T}]$;

(4)

Assume, in addition, that there are finitely many fixed points for the action of $\nu$. The category $\mathcal{O}_{\nu }(\mathcal{\overline{A}}_{\lambda }(n,\ell ))$ is highest weight with standard objects $\triangle _{\nu }(p_{i})$ and costandard objects $\nabla _{\nu }(p_{i})$ for $p_{i} \in \overline{\mathcal{M}}^{\theta }(n,\ell )^{T}$.

Remark 2.12.

The order required for highest weight structure comes from the contraction order on the fixed points. This is the order, in which $p_i\preceq _{\nu } p_j$ iff $p_i \in \overline{X^{\nu }_{p_j}}$, where $X^{\nu }_{p_j}\coloneq \{x\in \overline{\mathcal{M}}^{\theta }(n,\ell )\nobreakspace |\nobreakspace \lim \limits _{t\rightarrow 0}\nu (t)x=p_i\}$.

2.2. $T$-fixed points

To study the category $\mathcal{O}_{\nu }(\mathcal{\overline{A}}_{\lambda }(n,\ell ))$, we first need to obtain some information on the torus fixed points. This is summarized in Theorem 2.14.

Remark 2.13.

Since the case $\ell =1$ was studied in 20, henceforth we assume $\ell \geq 2$.

Theorem 2.14.

The variety $\overline{\mathcal{M}}^{\theta }(n,\ell )$ has finitely many $T$-fixed points if dim$V\leq 3$.

Proof.

Let $\tilde{p}=(X_{1}, \mathinner{\ldotp \ldotp \ldotp }, X_{\ell },Y_{1}, \mathinner{\ldotp \ldotp \ldotp }, Y_{\ell },i,j)\in \mu ^{-1}(0)$ be a point in the preimage of a fixed point $p\in \overline{\mathcal{M}}^{\theta }(n,\ell )$, then there exists a homomorphism $\eta _p:T \rightarrow G$, s.t. the following system of equalities is satisfied ($t=(t_{1},\mathinner{\ldotp \ldotp \ldotp }t_{\ell })\in T$):

$$\begin{equation} \begin{cases} t_{1}X_{1}=\eta _p(t)X_{1}\eta _p(t)^{-1}, \\ \mathinner{\ldotp \ldotp \ldotp }\\ t_{\ell }X_{\ell }=\eta _p(t)X_{\ell }\eta _p(t)^{-1}, \\ t_{1}^{-1}Y_{1}=\eta _p(t)Y_{1}\eta _p(t)^{-1}, \\ \mathinner{\ldotp \ldotp \ldotp }\\ t_{\ell }^{-1}Y_{\ell }=\eta _p(t)Y_{\ell }\eta _p(t)^{-1},\\ i=\eta _p(t)^{-1}i, \\ j=\eta _p(t)j. \end{cases} {\tag{7}} \end{equation}$$

Let $\{\varepsilon _1,\ldots ,\varepsilon _\ell \}$ be the set of coordinate characters of the torus $T$, i.e. $\varepsilon _i(t_1,\mathinner{\ldotp \ldotp \ldotp },t_\ell ) =t_i$. The weight decomposition of $V$ with respect to $\eta _p$ is

$$\begin{equation*} V=\underset{\chi \in char(T)}{\bigoplus } V_{\chi }, \end{equation*}$$

with $V_{\chi }=\{v\in V|\nobreakspace \eta _p(t)\cdot v=\chi (t)v\}$. It follows from the system of equation 7 that $X_i(V_\chi )\subset V_{\chi -\varepsilon _i}$ and, similarly, $Y_i(V_\chi )\subset V_{\chi +\varepsilon _i}$ (here multiplication of characters is written additively). As $im\nobreakspace j\neq 0$ due to the stability condition it follows from the last equation in 7 that im$\nobreakspace j\in V_0$.

Below we provide a description of the fixed points when dim$(V)\leq 3$.

Case 1.

If dim$(V)=1$, the variety $\overline{\mathcal{M}}^{\theta }(1,\ell )$ is a single point.

Case 2.

If dim$(V)=2$, we choose a cyclic vector $0\neq v_{0}\in im(j)$ (by $v$ being a cyclic vector we mean that $span_{\mathbb{C}}(f(X_1,\mathinner{\ldotp \ldotp \ldotp },X_{\ell },Y_1,\mathinner{\ldotp \ldotp \ldotp },Y_{\ell })v)_{f\in \mathbb{C}[t_1,\mathinner{\ldotp \ldotp \ldotp },t_{2\ell }]}=V$) as the first vector in the basis. Then at least one of the $X_{k}$ or $Y_{s}$ must act nontrivially on $v_{0}$ and the image is $v_{1}$ inside some $V_{\pm \varepsilon _i}$. The vectors $v_{0}$ and $v_{1}$ already span $V$ as they have different weights and cannot be collinear. We notice that $X_{s}v_{0}=v_{1}$ or $Y_{s}v_{0}=v_{1}$ immediately implies $X_{\neq s}v_{0}=Y_{\neq s}v_{0}=X_{\neq s}v_{1}=Y_{\neq s}v_{1}=0$ as all these vectors would lie in weight spaces different from $V_{0,\mathinner{\ldotp \ldotp \ldotp },0}$ and $V_{0,\mathinner{\ldotp \ldotp \ldotp },0,\pm 1_{s},0,\mathinner{\ldotp \ldotp \ldotp },0}$. It remains to notice that equation 6 becomes $[X_{s},Y_{s}]+ji=0$, which shows that $X_{s}\neq 0$ implies $Y_{s}=0$ and vice versa. Therefore, there are $2\ell$ fixed points: $p_{s}=(X_{\neq s}=0, X_{s}=\begin{pmatrix} 0 & 0 \\ 1 & 0\end{pmatrix}, Y_{1}=0, \mathinner{\ldotp \ldotp \ldotp }, Y_{\ell }=0, i=0, j=\begin{pmatrix} 1 \\ 0\end{pmatrix})$, $p_{s+\ell }=(X_{1}=0, \mathinner{\ldotp \ldotp \ldotp }, X_{\ell }=0, Y_{\neq s}=0, Y_{s}=\begin{pmatrix} 0 & 0 \\ 1 & 0\end{pmatrix}, i=0, j=\begin{pmatrix} 1 \\ 0\end{pmatrix})$ , where $s \in \{1,\mathinner{\ldotp \ldotp \ldotp }, \ell \}$.

Case 3.

Now dim$(V)=3$. Again let the cyclic vector $0\neq v_{0}\in im\nobreakspace j$ be the first vector in the basis. Now there are the following possibilities ($s,k \in \{1,\mathinner{\ldotp \ldotp \ldotp },\ell \}$):

• for some $s,k$: $X_{s}v_{0}=v_{1}\neq 0$ and $Y_{k}v_{0}=v_{2}\neq 0$;

• for some $s\neq k$: $X_{s}v_{0}=v_{1}\neq 0$ and $X_{k}v_{0}=v_{2}\neq 0$;

• for some $s\neq k$: $Y_{s}v_{0}=v_{1}\neq 0$ and $Y_{k}v_{0}=v_{2}\neq 0$;

• for some $s\neq k$: $X_{s}v_{0}=v_{1}\neq 0$ and $Y_{k}v_{1}=v_{2}\neq 0$;

• for some $s,k$: $X_{s}v_{0}=v_{1}\neq 0$ and $X_{k}v_{1}=v_{2}\neq 0$;

• for some $s,k$: $Y_{s}v_{0}=v_{1}\neq 0$ and $Y_{k}v_{1}=v_{2}\neq 0$;

In each of the cases above the vectors $v_{0}, v_{1}$ and $v_{2}$ are linearly independent and span $V$, while all the remaining $X$ and $Y$ coordinates of $p$ are zero. We verify it when $X_{s}v_{0}=v_{1}$ and $Y_{k}v_{0}=v_{2}$, the remaining cases being similar.

First, $X_{\neq k}$ and $Y_{\neq s}$ must be zero, as otherwise there would be vectors with weights different from those of $v_{0}, v_{1}$ and $v_{2}$ and, therefore, linearly independent with them. For the same reason $X_{k}v_{0}=X_{k}v_{1}=X_{s}v_{1}=X_{s}v_{2}=Y_{s}v_{0}=Y_{s}v_{2}=Y_{k}v_{1}=Y_{k}v_{2}=0$. To show $Y_{s}v_{1}=0$, we notice that equation 6 reduces to $[X_{s}, Y_{s}]+[X_{k}, Y_{k}] + ji = 0$. Applying to $v_{1}$, we get

$$\begin{equation*} X_{s}Y_{s}v_{1}+jiv_{1}=0, \end{equation*}$$

and notice that $X_{s}Y_{s}v_{1} \in V_{0,\mathinner{\ldotp \ldotp \ldotp },-1_{s},\mathinner{\ldotp \ldotp \ldotp },0}$, while $ji v_{1} \in V_{0,\mathinner{\ldotp \ldotp \ldotp },0_{s},\mathinner{\ldotp \ldotp \ldotp },0}$. Thus, $jiv_{1}=0$ and $X_{s}Y_{s}v_{1}=0$ separately, so $Y_{s}v_{1}=0$ and $Y_{s}=0$. It is analogous to show that $X_{k}=0$.■

Remark 2.15.

Next we show that when $n=4, \ell =2$ the subvariety of fixed points contains a copy of the projective line $\mathbb{CP}^{1}=\mathbb{P}(span_\mathbb{C}(\mu _{1},\mu _{2}))$. The operators below are presented in a weight basis with the first vector of weight $(0,0)$, the second $(-1,0)$, the third $(0,-1)$ and the fourth $(-1,-1)$, the action of the subgroup of $G$, preserving the weight decomposition, can only simultaneously rescale $\mu _{1}$ and $\mu _{2}$. The subvariety is given by

$$\begin{equation*} X_{1}=\begin{pmatrix} 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & \mu _{1} & 0 \end{pmatrix},\nobreakspace X_{2}=\begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & \mu _{2} & 0 & 0 \end{pmatrix},\nobreakspace Y_{1}=Y_{2}=0,\nobreakspace i=0,\nobreakspace j=\begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix}. \end{equation*}$$

Remark 2.16.

Both varieties $\overline{\mathcal{M}}^{\theta }(1,\ell )$ and $\overline{\mathcal{M}}(1,\ell )$ consist of a single point, therefore, we proceed with the case $dim V=2$.

The following fact is a particular case of the result established in Section $5$ of 21 and will be used in the proof of Theorem 6.15. Suppose $\tilde{\nu }$ lies in the face of a chamber containing $\nu$. Then $\triangle _{\tilde{\nu }}$ restricts to an exact functor $\mathcal{O}_{\nu }(C_{\tilde{\nu }}(\mathcal{\overline{A}}_{\lambda }(2,\ell ))) \rightarrow \mathcal{O}_{\nu }(\mathcal{\overline{A}}_{\lambda }(2,\ell ))$. Moreover, there is an isomorphism of functors $\triangle _{\nu }=\triangle _{\tilde{\nu }}\circ \underline{\triangle }$, where $\triangle _{\tilde{\nu }}: C_{\tilde{\nu }}(\mathcal{A}_{\lambda })\operatorname {-mod} \rightarrow \mathcal{A}_{\lambda }\operatorname {-mod}, \underline{\triangle }: C_{\nu }(\mathcal{A}_{\lambda })\operatorname {-mod} \rightarrow C_{\tilde{\nu }}(\mathcal{A}_{\lambda })\operatorname {-mod}$ and $\triangle _{\nu }$ is the standardization functor given by Definition 2.8. This allows to study the functor $\triangle _{\tilde{\nu }}$ in stages.

We start by describing the fixed points loci $\overline{\mathcal{M}}^{\theta }(2, \ell )^{\nu (\mathbb{C}^{*})}$ for certain one-parameter subgroups $\tilde{\nu }: \mathbb{C}^{*} \rightarrow T$ and the corresponding algebras $C_{\tilde{\nu }}(\mathcal{A}_{\lambda })$.

Theorem 2.17.

The fixed point set $\overline{\mathcal{M}}^{\theta }(2, \ell )^{\tilde{\nu }(\mathbb{C}^{*})}$ for $\tilde{\nu }: \mathbb{C}^{*}\rightarrow T$ with $\tilde{\nu }(t)=(t^d,1,\mathinner{\ldotp \ldotp \ldotp },1)$ and $d \in \mathbb{Z}_{>0}$ is $\overline{\mathcal{M}}^{\theta }(2, \ell -1)\amalg \mathbb{C}^{2\ell -2}\amalg \mathbb{C}^{2\ell -2}$.

Proof.

The subset $\overline{\mathcal{M}}^{\theta }(2,\ell )^{\tilde{\nu }(\mathbb{C}^{*})}$ is formed by the points $p=(X_{1},\mathinner{\ldotp \ldotp \ldotp },X_{\ell },Y_{1},\mathinner{\ldotp \ldotp \ldotp }, Y_{\ell },i,j)$ which satisfy the system of equation 8. These equations are obtained analogously to those in 7 with $\eta _p$ standing for the composition $\mathbb{C}^*\stackrel{\tilde{\nu }}{\rightarrow }T \rightarrow G$, s.t.

$$\begin{equation} \begin{cases} t^{d}X_{1}=\eta _p(t)X_{1}\eta _p(t)^{-1}, \\ X_{2}=\eta _p(t)X_{2}\eta _p(t)^{-1}, \\ \mathinner{\ldotp \ldotp \ldotp }\\ X_{\ell }=\eta _p(t)X_{\ell }\eta _p(t)^{-1}, \\ t^{-d}Y_{1}=\eta _p(t)Y_{1}\eta _p(t)^{-1}, \\ Y_{2}=\eta _p(t)Y_{2}\eta _p(t)^{-1}, \\ \mathinner{\ldotp \ldotp \ldotp }\\ Y_{\ell }=\eta _p(t)Y_{\ell }\eta _p(t)^{-1},\\ i=\eta _p(t)^{-1}i, \\ j=\eta _p(t)j, \end{cases} {\tag{8}} \end{equation}$$

and $\eta _p$ is the same for points in the same connected component.

There are two possible cases. First, if $X_1=Y_1=0$, it follows from 8 and our choice of the stability condition that the entire $2$-dimensional vector space $V$ is of weight $0$ with respect to $\eta _p(t)$ and, thus, $\eta _p(t)=\begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix}$. Such points form the fixed component $\overline{\mathcal{M}}^{\theta }(2, \ell -1)\subset \overline{\mathcal{M}}^{\theta }(2,\ell )^{\tilde{\nu }(\mathbb{C}^{*})}$.

Next we treat the case when $(X_1,Y_1)\neq 0$. Let $v_0\in im\nobreakspace j$ be a cyclic vector. Notice that since dim$(V)=2$ and $X_1 v_0\subset V_{-d\varepsilon _1}$ while $Y_1 v_0\subset V_{d\varepsilon _1}$, we must have that at least one of the operators $X_1,Y_1$ is zero as well as the remaining one squared. Therefore, the matrix of the nonzero operator is conjugate to $\begin{pmatrix} 0 & 0 \\ 1 & 0\end{pmatrix}$. One observes that $X_1=\begin{pmatrix} 0 & 0 \\ 1 & 0\end{pmatrix}$ and $Y_1=0$ implies the weight basis of $V$ consists of vectors with weights $0$ and $d$, while $\eta _p(t)=\begin{pmatrix} 1 & 0 \\ 0 & t^{d}\end{pmatrix}$ in this basis, similarly, $\eta _p(t)=\begin{pmatrix} 1 & 0 \\ 0 & t^{-d}\end{pmatrix}$ if $Y_1=\begin{pmatrix} 0 & 0 \\ 1 & 0\end{pmatrix}$ and $X_1=0$. In either of the two cases a $\tilde{\nu }(\mathbb{C}^{*})$-fixed point must have $\{X_2,\mathinner{\ldotp \ldotp \ldotp },X_{\ell },Y_2,\mathinner{\ldotp \ldotp \ldotp },Y_{\ell }\nobreakspace |\nobreakspace X_i,Y_j\in \mathfrak{h}\subset \mathfrak{sl}_2\}$ and none of such points are identified under the action of $G$, hence, we arrive with an irreducible component isomorphic to $\mathbb{C}^{2\ell -2}$.

■

Remark 2.18.

Let $T'\simeq (\mathbb{C}^{*})^{\ell -1}\coloneq \{(t_1,\mathinner{\ldotp \ldotp \ldotp },t_{\ell -1},t_{\ell })\subset T\nobreakspace |\nobreakspace t_{\ell }=1\}$, then the irreducible components of $\overline{\mathcal{M}}^{\theta }(2,\ell )^{T'}$ are $\{p\in \overline{\mathcal{M}}^{\theta }(2,\ell )\nobreakspace |\nobreakspace X_1=\mathinner{\ldotp \ldotp \ldotp }=X_{\ell -1}=Y_1=\mathinner{\ldotp \ldotp \ldotp }=Y_{\ell -1}=0\}\simeq T^{*}\mathbb{P}^{1}$ and $2\ell -2$ copies of $\mathbb{C}^{2}$. Indeed, now $X_{\ell }$ and $Y_{\ell }$ preserve the weights of the weight vectors. Therefore, there are two possibilities:

(i)

the vector space $V=V_0$, so $X_{\neq \ell }=Y_{\neq \ell }=0$ and we arrive at $T^{*}\mathbb{P}^{1}$ described above;

(ii)

$V$ is spanned by $v_{0}\in V_0$ and $v_1\in V_{\pm \varepsilon _s}$, in which case $X_{\ell }=\begin{pmatrix} a & 0 \\ 0 & -a\end{pmatrix}, Y_{\ell }=\begin{pmatrix} b & 0 \\ 0 & -b\end{pmatrix}$, one of $X_{s}$ or $Y_{s}$ is $\begin{pmatrix} 0 & 0 \\ 1 & 0\end{pmatrix}$ (depending on the sign of the corresponding weight of $v_{1}$), the other $X$’s and $Y$’s as well as $i$ are $0$ and $j=\begin{pmatrix} 1 \\ 0 \end{pmatrix}$. Since the remaining action of $G$ is trivial and $s\in \{1,\mathinner{\ldotp \ldotp \ldotp },\ell -1\}$, this gives rise to $2\ell -2$ copies of $\mathbb{C}^{2}$.

Proposition 2.19.

Let $\nu _0$ and $\nu '$ be the one-parameter subgroups from Theorem 2.17.

(a)

We have an isomorphism of algebras $C_{\nu _0}(\overline{\mathcal{A}}_{\lambda }(2,\ell ))\simeq \overline{\mathcal{A}}_{\lambda }(2,\ell -1)\oplus \mathcal{D}(\mathbb{C}^{2\ell -2})\oplus \mathcal{D}(\mathbb{C}^{2\ell -2})$, where $\overline{\mathcal{A}}_{\lambda }(2,\ell -1)$ is a quantization of $Z=\overline{\mathcal{M}}^{\theta }(2,\ell -1)$.

(b)

Similarly, $C_{\nu '}(\overline{\mathcal{A}}_{\lambda }(2,\ell ))\simeq \mathcal{A}^{Z_1}_{\lambda +1-\frac{\ell }{2}}\oplus \mathcal{A}^{Z_2}_{\lambda +\frac{\ell }{2}}$, where $Z_1, Z_2$ are the fixed components for $\nu '$ and $\mathcal{A}^{Z_i}_{\mu }$ stands for the quantization of $Z_i$ with period $\mu$.

Proof.

An application of $(4)$, Proposition $2.2$ in 20 gives that $C_{\nu _0}(\overline{\mathcal{A}}^{sym}_{\lambda }(2,\ell ))=\underset{k}{\oplus }\mathcal{A}^{Z_k}_{i_{Z_k}^{*}(\lambda )-\rho _{Z_k}}$, where $Z_k$’s are the irreducible components of $\overline{\mathcal{M}}^{\theta }(2,\ell )^{\nu _0}$ and $\mathcal{A}^{Z_k}_{i_{Z_k}^{*}(\lambda )-\rho _{Z_k}}$ stands for the algebra of global sections of the filtered quantization of $Z_k$ with period ${i_{Z_k}^{*}(\lambda )-\rho _{Z_k}}$. Here $i_{Z}^{*}$ is the pull-back map $H^2(\overline{\mathcal{M}}^{\theta }(2,\ell ),\mathbb{C})\rightarrow H^2(Z,\mathbb{C})$ and $\rho _{Z_k}$ equals half of the first Chern class of the contracting bundle of $Z_k$. We start with describing this bundle in our case. For the general description of tangent spaces to quiver varieties we refer to Lemma $3.10$ and Corollary $3.12$ in 26. The tangent bundle descends from the $G$-module ker $\beta$ / im $\alpha$, where $\alpha$ and $\beta$ are in the following complex:

$$\begin{equation} \operatorname {Hom}(V,V) \stackrel{\alpha }{\hookrightarrow } \mathfrak{sl}_2\otimes \mathbb{C}^{2\ell }\oplus V\oplus V^* \overset{\beta }{\twoheadrightarrow } \operatorname {Hom}(V,V), {\tag{9}} \end{equation}$$

here $\alpha$ stands for the differential of the $G$-action and $\beta$ is the differential of the moment map at that fixed point.

It is not hard to observe that the sequence 9 is equivariant with respect to the $(\mathbb{C}^{*})^{\ell }$-action with $\beta$ surjective and $\alpha$ injective.

We proceed with verifying the assertion of (a). As every bundle over the $\mathbb{C}^{2\ell -2}$ component of $Z$ is trivial, we look at the restriction of the contracting bundle to $\overline{\mathcal{M}}^{\theta }(2,\ell -1)$.

It follows from the description of the tangent bundle as the middle cohomology of the complex 9 that the contracting bundle descends under $G$-action from $T^*\overline{R}^{\tilde{\eta }_p, >0}$ modulo two copies of $\mathfrak{g}^{\tilde{\eta }_p, >0}$. In our case $(T^*\overline{R})^{\tilde{\eta }_p, >0}=H$ is the three-dimensional space $Vec(X_1)$, while $\mathfrak{g}$ is pointwise fixed under the action of $\tilde{\eta }_p$, hence, the contracting bundle descends from $H$.

The top exterior power of the vector bundle $\widetilde{H}$ descending from $H$ under $G$-action is trivial, since $G$ acts trivially on the top exterior power of $H$. By 17, Section 5, the period of a quantization $\overline{\mathcal{A}}_{\lambda }(2,\ell )$ is $\lambda -\zeta$, where $\zeta$ is half the character of the action of $G$ on $\Lambda ^{top}\overline{R}$. Thus the periods of the quantizations $\overline{\mathcal{A}}_{\lambda }(2,\ell )$ and $\overline{\mathcal{A}}_{\lambda }(2,\ell -1)$ are both equal to $\lambda +\frac{1}{2}$, the first claim of the proposition follows.

We verify the claim in $(b)$ for $Z_1$. There is a line subbundle $L_{triv} \subset \tilde{V}$ with the fiber over a point $p\in Z_1$ being $\operatorname {im}j$. It is trivial, since for a fixed $0\neq w\in W$ one has a nowhere vanishing section $j(w)$ of $\tilde{V}$. Using the splitting principle, we write $\tilde{V}=L_{triv}\oplus L_1$ with $c_1(L_{triv})=0$ and $c_1(L_1)=c_{Z_1}$, where $c_{Z_1}$ is the generator of $H^2(Z_1)$. In this case $V=\mathbb{C}\langle v_0,v_1\rangle$ with $v_0$ of weight $0$ and $v_1$ of weight $d$, in other words, $\eta _p=\begin{pmatrix} 1 & 0 \\ 0 & t^d \end{pmatrix}$ in the basis $\langle v_0, v_1\rangle$. This implies that the bundle on $Z_1$ descending from $\mathfrak{sl}_2$ is $L_{triv}\oplus L_1\oplus L_1^*$. Let $\tilde{\eta }_p(t)=(\nu '(t),\eta _p(t))\subset T\times G$, then $U^{\tilde{\eta }_p, >0}=(z_1,\mathinner{\ldotp \ldotp \ldotp }, z_{\ell },v_1)$, where $z_s$ is the $12$-entry (first row and second column) of the matrix $X_s$, while $\mathfrak{g}^{\tilde{\eta }_p, >0}$ consists of the $12$-entry of the corresponding matrix. Hence, the nontrivial part of the contracting bundle is $L_1\otimes \mathbb{C}^{\ell -1}$. Thus we conclude that $\rho _{Z_1}=\frac{\ell -1}{2}c_{Z_1}$.

Analogously one can show that the nontrivial part of the contracting bundle on $Z_2$ is $L_1^*\otimes \mathbb{C}^{\ell -1}$ and $\rho _{Z_2}=\frac{1-\ell }{2}c_{Z_2}$. The maps $i_{Z_1}^{*}$ and $i_{Z_2}^{*}$ send $c_1(\tilde{V})\in H^2(\overline{\mathcal{M}}^{\theta }(2,\ell ))$ to the generators $c_{Z_1}\in H^2(Z_1)$ and $c_{Z_2}\in H^2(Z_2)$. The claim in (b) follows.

■

Remark 2.20.

The quantizations $\mathcal{A}^{Z_1}_{\lambda +1-\frac{\ell }{2}}\text{ and }\mathcal{A}^{Z_2}_{\lambda +\frac{\ell }{2}}$ are isomorphic to $\mathcal{D}^{\lambda -\ell +1}(\mathbb{P}^{\ell -1})$ and $\mathcal{D}^{\lambda }(\mathbb{P}^{\ell -1})$ (the algebras of twisted differential operators on projective spaces).

2.3. Central fibers

Lemma 2.21 provides some information on the preimages of zero under $\bar{\rho }:\overline{\mathcal{M}}^{\theta }(n,\ell )\rightarrow \overline{\mathcal{M}}(n,\ell )$ (central fibers) in $\overline{\mathcal{M}}^{\theta }(n,\ell )$.

Lemma 2.21.

(a)

The preimage of $0$ in $\overline{\mathcal{M}}^{\theta }(2,\ell )$ is $\bar{\rho }^{-1}(0)=\mathbb{P}^{2\ell -1}$.

(b)

Let $n,\ell >1$, then $\operatorname {dim}(\bar{\rho }^{-1}(0))<\frac{1}{2}\operatorname {dim}(\overline{\mathcal{M}}^{\theta }(n,\ell ))$.

Proof.

An application of the Hilbert-Mumford criterion shows (the argument is analogous to the one in Proposition $9.7.4$. in 11) that $p\in \overline{\mathcal{M}}^{\theta }(n,\ell )$ lies in $\bar{\rho }^{-1}(0)$ if and only if on the corresponding representation there exists a filtration $0 = L_0 \subset L_1 \subset L_2 \subset \mathinner{\ldotp \ldotp \ldotp }\subset L_n = r_p \in T^*\overline{R}$ by subrepresentations such that each quotient $L_i/L_{i-1}$ for $i<n$ is isomorphic to a simple representation (of the framed quiver $\widehat{B}_{2\ell }$) with dimension vector $\begin{pmatrix} 0 \\ 1\end{pmatrix}$ and $L_n/L_{n-1}$ is isomorphic to the simple representation with dimension vector $\begin{pmatrix} 1 \\ 0\end{pmatrix}$ (here the top coordinate corresponds to the dimension of framing and the bottom to the dimension of $V$). This implies that all the $\mathfrak{sl}_n$-components of $p$ must be strictly upper-triangular matrices. It follows from equation 6 and our choice of stability condition that $i=0$.

(a) Pick a vector $0\neq h \in im\nobreakspace j$. As $h$ is a cyclic vector, it must have a nontrivial projection onto $L/L_1$. The action by matrices of the form $\begin{pmatrix} 1 & \alpha \\ 0 & 1\end{pmatrix}$ (conjugation by which does not change any of the $2\times 2$ matrices of $p$) allows to assume that the component of $h$ along the first vector is zero. Acting by $\begin{pmatrix} 1 & 0 \\ 0 & *\end{pmatrix} \subset GL_{2}$ allows to pick a representative of $p$ with $j=\begin{pmatrix} 0 \\ 1\end{pmatrix}$ and the action by $\begin{pmatrix} * & 0 \\ 0 & 1\end{pmatrix} \subset GL_{2}$ to simultaneously rescale the $2\times 2$ matrices of $p$. We conclude that $p=(X_{1}=\begin{pmatrix} 0 & a_{1} \\ 0 & 0\end{pmatrix}, \mathinner{\ldotp \ldotp \ldotp }, X_{\ell }=\begin{pmatrix} 0 & a_{\ell } \\ 0 & 0\end{pmatrix},Y_{1}=\begin{pmatrix} 0 & a_{\ell +1} \\ 0 & 0\end{pmatrix}, \mathinner{\ldotp \ldotp \ldotp }, Y_{\ell }=\begin{pmatrix} 0 & a_{2\ell } \\ 0 & 0\end{pmatrix}, i=0, j=\begin{pmatrix} 0 \\ 1\end{pmatrix})$ with at least one of $X_{k}$’s and $Y_{s}$’s, being nonzero due to the stability condition, up to simultaneous dilations of $X_{k}$’s and $Y_{s}$’s, which shows the claim, stated in (a).

Now we show the claim in (b). Acting by matrices of the form

$$\begin{equation*} \begin{pmatrix} 1 & 0 & \mathinner{\ldotp \ldotp \ldotp }& 0 & * \\ 0 & 1 & \mathinner{\ldotp \ldotp \ldotp }& 0 & * \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \mathinner{\ldotp \ldotp \ldotp }& 1 & * \\ 0 & 0 & \mathinner{\ldotp \ldotp \ldotp }& 0 & 1 \\ \end{pmatrix}, \end{equation*}$$

we can assume that $h$ is proportional to the last vector in the basis. The action by the subgroup

$$\begin{equation*} \begin{pmatrix} 1 & 0 & \mathinner{\ldotp \ldotp \ldotp }& 0 & 0 \\ 0 & 1 & \mathinner{\ldotp \ldotp \ldotp }& 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \mathinner{\ldotp \ldotp \ldotp }& 1 & 0 \\ 0 & 0 & \mathinner{\ldotp \ldotp \ldotp }& 0 & * \\ \end{pmatrix}\subset GL_{n} \end{equation*}$$

allows to assume $j=\begin{pmatrix} 0 \\ \vdots \\ 0 \\ 1\end{pmatrix}$.

Since $i=0$, the moment equation 6 reduces to $\sum \limits _{k=0}^{\ell }[X_k,Y_k]=0$ and as each of the commutators is a matrix of the form

$$\begin{equation*} [X_k,Y_k]=\begin{pmatrix} 0 & 0 & * & * & \mathinner{\ldotp \ldotp \ldotp }& * \\ 0 & 0 & 0 & * & \mathinner{\ldotp \ldotp \ldotp }& * \\ \vdots & \vdots & \vdots & \ddots & \ddots & \vdots \\ 0 & 0 & 0 & 0 & \mathinner{\ldotp \ldotp \ldotp }& * \\ 0 & 0 & 0 & 0 & \mathinner{\ldotp \ldotp \ldotp }& 0 \\ \end{pmatrix}, \end{equation*}$$

equation 6 imposes $\frac{(n-1)(n-2)}{2}$ independent conditions on the coordinates of $p\in \bar{\rho }^{-1}(0)$. The action of matrices of the form

$$\begin{equation*} \begin{pmatrix} * & * & \mathinner{\ldotp \ldotp \ldotp }& * & 0 \\ 0 & * & \mathinner{\ldotp \ldotp \ldotp }& * & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \mathinner{\ldotp \ldotp \ldotp }& * & 0 \\ 0 & 0 & \mathinner{\ldotp \ldotp \ldotp }& 0 & 1 \\ \end{pmatrix} \end{equation*}$$

preserves both $j$ and the strictly upper-triangular matrices and reduces the dimension by $\frac{n(n-1)}{2}$. Therefore, we have established that

$$\begin{equation*} \operatorname {dim}(\bar{\rho }^{-1}(0))\leq \frac{n(n-1)}{2}2\ell -\frac{(n-1)(n-2)}{2}-\frac{n(n-1)}{2}=(n^2-n)\ell -n^2+2n-1 \end{equation*}$$

and a straightforward computation shows that $(n^2-n)\ell -n^2+2n-1<(\ell -1)n^2-\ell +n=\frac{1}{2}\operatorname {dim}(\overline{\mathcal{M}}^{\theta }(n,\ell ))$ provided $n,\ell >1$.

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Corollary 2.22.

Assume $n,\ell >1$, then the central fiber $\bar{\rho }^{-1}(0)\subset \overline{\mathcal{M}}^{\theta }(n,\ell )$ is an isotropic but not Lagrangian subvariety.

Remark 2.23.

The $T$-fixed points for the action on $\overline{\mathcal{M}}^{\theta }(2,\ell )$ (see Theorem 2.14) lie on $\overline{\rho }^{-1}(0)=\mathbb{P}^{2\ell -1}$.

Corollary 2.24.

$H^2(\overline{\mathcal{M}}^{\theta }(2,\ell ))\simeq \mathbb{C}$.

Proof.

This follows from the fact that $\overline{\mathcal{M}}^{\theta }(2,\ell )$ is homotopy equivalent to the central fiber, while the latter is isomorphic to $\mathbb{P}^{2\ell -1}$ as shown in Lemma 2.21(a).

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Corollary 2.25.

There are no finite dimensional $\overline{\mathcal{A}}_{\lambda }(n,\ell )$-modules for $n, \ell >1$ and generic $\nu$.

Proof.

The support of a finite dimensional module $M$ must be $0 \in \overline{\mathcal{M}}(n,\ell )$ (since $0$ is the only fixed point of $\overline{\mathcal{M}}(n,\ell )$ for the $\mathbb{S}$-action, the support is $\mathbb{S}$-stable and the module is finite dimensional). Notice that the support of $Loc(M)$ is contained in $\bar{\rho }^{-1}(0) \subset \overline{\mathcal{M}}^{\theta }(n,\ell )$. On the other hand, due to Gabber’s involutivity theorem (see 14), the support of a coherent module must be a coisotropic subvariety of $\overline{\mathcal{M}}^{\theta }(n,\ell )$. However, this is impossible for dimension reasons.

■

I would like to thank Pavel Etingof and Ivan Losev for bringing my attention to the following fact.

Let $A$ be a Poisson algebra over $\mathbb{C}$, i.e. $A = \mathcal{O}(X)$, where X is an affine Poisson variety.

Definition 2.26.

The zeroth Poisson homology, $HP_0(A)$ is the quotient $A/\{A, A\}$.

Conjecture 2.27 was formulated in 12 (see Conjecture $1.3.1$ therein).

Conjecture 2.27.

Let $\rho : \tilde{X}\rightarrow X$ be a symplectic resolution with $X$ affine, then $HP_0(\mathcal{O}(X))= H^{dim X}(\tilde{X})$.

Conjecture 2.27 holds in many cases (see Examples $6.4-6.7$ in 13 for details):

(1)

Let $Y$ be a smooth symplectic surface. Set $X = Sym^n Y \coloneq Y^n/S_n$, the $n$-th symmetric power of $Y$ and consider the resolution $\bar{\rho }: \tilde{X} = Hilb^n Y \rightarrow X$.

(2)

Take $Y = \mathbb{C}^2/\Gamma$ and the crepant resolution $\tilde{Y}\rightarrow Y$ (here $\Gamma \subset SL(2, \mathbb{C})$ is a finite subgroup), consider $X\coloneq Sym^n Y$ and the resolution $\rho _1 : \tilde{X} \coloneq Hilb^n \tilde{Y} \rightarrow Sym^n \tilde{Y}$. Now compose this with $\rho _2 : Sym^n \tilde{Y} \rightarrow Sym^n Y$ to obtain the resolution $\rho = \rho _2 \circ \rho _1: \tilde{X} \rightarrow X$.

(3)

Let $\mathcal{N}$ be the cone of nilpotent elements in a complex semisimple Lie algebra $\mathfrak{g}$, and $\rho$ the Springer resolution $T^*(G/B)\rightarrow \mathcal{N}$.

Remark 2.28.

The resolutions $\overline{\rho }:\overline{\mathcal{M}}^{\theta }(n,\ell )\rightarrow \overline{\mathcal{M}}(n,\ell )$ serve as counterexamples to Conjecture 2.27. Indeed, $H^{top}(\overline{\mathcal{M}}^{\theta }(2,\ell ),\mathbb{C})=H^{3\ell -2}(\mathbb{P}^{2\ell -1},\mathbb{C})=0$ and, in general, $H^{top}(\overline{\mathcal{M}}^{\theta }(n,\ell ),\mathbb{C})=H^{\frac{1}{2}\operatorname {dim}(\overline{\mathcal{M}}^{\theta }(n,\ell ))}(\overline{\mathcal{M}}^{\theta }(n,\ell ),\mathbb{C})=0$, since the variety $\overline{\mathcal{M}}^{\theta }(n,\ell )$ is homotopy equivalent to $\overline{\rho }^{-1}(0)$ (via the contracting $\mathbb{C}^*$-action) and this variety has dimension strictly less than $\frac{1}{2}\operatorname {dim}(\overline{\mathcal{M}}^{\theta }(n,\ell ))$ as shown in Lemma 2.21(b). On the other hand, the point $0$ is a symplectic leaf in affine Poisson varieties $\overline{\mathcal{M}}(n,\ell )$. This is true, since the Poisson bracket is of degree $-2$ and there are no invariant functions of degree one in $\mathbb{C}[T^*\bar{R}]^G$, hence, the maximal ideal of $0$ is Poisson. From this it follows that the vector spaces $HP_0(\mathcal{O}(\overline{\mathcal{M}}(n,\ell )))$ are at least $1-$dimensional. Therefore, $H^{top}(\overline{\mathcal{M}}(n,\ell ))\neq HP_0(\overline{\mathcal{M}}(n,\ell ))$, contradicting the claim of Conjecture 2.27.

3. Symplectic leaves and slices

3.1. Symplectic leaves

First we describe the symplectic leaves and slices to them for the Poisson varieties $\overline{\mathcal{M}}(2,\ell )$ and $\overline{\mathcal{M}}(3,\ell )$. The general description was given by Nakajima, it can also be found in Section $2$ of 5. In particular (Section $6$ of 25 or Section $3$ of 26), it was shown that

$$\begin{equation*} \overline{\mathcal{M}}(n,\ell )=\underset{\hat{G}\subseteq G}{\bigcup }\overline{\mathcal{M}}(n,\ell )_{\hat{G}}, \end{equation*}$$

where the strata are parametrized by reductive subgroups $\hat{G}\subseteq G$ and $\overline{\mathcal{M}}(n,\ell )_{\hat{G}}$ stands for the locus of isomorphism classes of semisimple representations, whose stabilizer is conjugate to $\hat{G}$. A semisimple representation $r \in T^{*}R$ is in $\overline{\mathcal{M}}(n,\ell )_{\hat{G}}$, if it can be decomposed as $r=r^{0}\oplus \underset{i=1}{\overset{n}{\bigoplus }} (r^{i}\otimes U_{i})$, where $r^{i}$’s for $i>0$ are simple and pairwise nonisomorphic with zero-dimensional framing and $U_{i}$’s are their multiplicity spaces, and $\hat{G}$ is conjugate to $\prod GL(U_{i})$. Moreover, according to Theorem $1.3$ of 9, the stratum $\overline{\mathcal{M}}(n,\ell )_{\hat{G}}$ is an irreducible locally closed subset of $\overline{\mathcal{M}}(n,\ell )$. Each stratum $\overline{\mathcal{M}}(n,\ell )_{\hat{G}}$, being irreducible, must be a symplectic leaf. The information about the symplectic leaves of $\overline{\mathcal{M}}(2,\ell )$ and $\overline{\mathcal{M}}(3,\ell )$ is summarized in Tables 1 and 2.

Remark 3.1.

We would like to notice that there are no irreducible representations with dimension vector $(1,1)$, as each summand $[X_{k},Y_{k}]$ in equation 6 equals zero and, therefore, $ji=0$ as well, forcing $i=0$ or $j=0$ (or $i=j=0$) and making the representation with dimension vector $(1,0)$ (zero-dimensional framing) in the former case and with dimension vector $(0,1)$ in the latter a subrepresentation.

The third leaf in Table 1 corresponds to representations $r=r^0\oplus r^1\oplus r^2$, while the fourth $r=r^0\oplus r^1\otimes \mathbb{C}^2$, the multiplicities in Table 2 are indicated in the second column therein.

Remark 3.2.

Since $\overline{\mathcal{M}}(2,\ell )$ has a unique symplectic leaf of codimension $2$, the slice to which is an $A_1$ singularity the Namikawa Weyl group (see 28) of $\overline{\mathcal{M}}(2,\ell )$ is $\mathbb{Z}/2\mathbb{Z}$. As there are no symplectic leaves of codimension $2$ in $\overline{\mathcal{M}}(3,\ell )$, the corresponding Namikawa Weyl group is trivial.

3.2. Fixed points on the slice

Next we study the slice taken at some point of the leaf of type $3$ in Table 1. This slice is the quiver variety in Figure 1 with $k,s \in \{1,\mathinner{\ldotp \ldotp \ldotp },\ell -1,\ell +1,\mathinner{\ldotp \ldotp \ldotp },2\ell -1\}$. The dimension vector is $(1,1)$ and the framing is also one-dimensional.

We consider the point $p=(X_{\ell }=\begin{pmatrix} 1 & 0 \\ 0 & -1\end{pmatrix},X_{\neq \ell }=0, Y_{k}=0,i=0,j=0)$. As the representation is semisimple, the $G$ orbit through $p$ in $T^*\overline{R}$ is closed and slightly abusing notation we will refer to the corresponding point in $\overline{\mathcal{M}}(2,\ell )$ as $p$ as well. The slice to the symplectic leaf at $p$ will be denoted by $\mathcal{SL}_{p}$. The description of slices as quiver varieties can be found in Section $2$ of 5. In our case the slice $\mathcal{SL}_{p}$ is the hypertoric variety obtained from the $(\mathbb{C}^*)^2$-action on $\mathbb{C}^{2\ell }$. In the basis $\langle x_{1}, x_{2},\mathinner{\ldotp \ldotp \ldotp }, x_{2\ell -2}, i_{1}, i_{2}\rangle$ the weights are $(t_{1}^{-1}t_{2},\mathinner{\ldotp \ldotp \ldotp },t_{1}^{-1}t_{2},t_{1}t_{2}^{-1},\mathinner{\ldotp \ldotp \ldotp },t_{1}t_{2}^{-1}, t_{1}^{-1},t_{2}^{-1})$. It is the quiver variety for the underlying graph depicted on Figure 1 with one-dimensional vector spaces assigned to the vertices and one-dimensional framing. We denote by $\rho _{s}$ the map $\mathcal{SL}_{p}^{\theta } \rightarrow \mathcal{SL}_{p}$ and fix $\theta =(-1,-1)$. The preimage of zero $\rho _{s}^{-1}(0)$ and the fixed points for the $T'\simeq (\mathbb{C}^{*})^{\ell -1}$-action on $\mathcal{SL}_{p}^{\theta }$ are described in Proposition 3.3.

Proposition 3.3.

(a)

$\rho _{s}^{-1}(0)\cong \mathbb{CP}^{2\ell -2}\cup\nobreakspace \mathbb{CP}^{2\ell -2}$ consists of two irreducible components, intersecting in a single point $(x_{s}=y_{k}=i_{1}=i_{2}=0, j_{1}=j_{2}=1)$.

(b)

There are $4\ell -3$ fixed points on $\mathcal{SL}_p^{\theta }$ for the $T'$-action. These points are (the $(\mathbb{C}^*)^2$-orbits of) $(x_{i}=1, x_{\neq i}=y_{j}=i_{1}=i_{2}=j_{2}=0, j_{1}=1)$, $(y_{j}=1, x_{i}=y_{\neq j}=i_{1}=i_{2}=j_{1}=0, j_{2}=1)$ and $(x_{s}=y_{k}=i_{1}=i_{2}=0, j_{1}=j_{2}=1)$.

Proof.

To see that (a) is true, we first notice that for $(\mathbf{x},\mathbf{y},i,j)\in \rho _s^{-1}(0)$ we have either all $x_{k}=0$ or all $y_{s}=0$ (use the Hilbert-Mumford criterion in a similar way to the proof of Lemma 2.21). In the former case the stability condition guarantees $j_{2}\neq 0$ and $j_{1}$ or at least one of $y_{s}$’s is nonzero. Therefore, the first equation in 10 immediately implies that $i_{2}=0$. To see that $i_{1}=0$ as well, notice that the one-dimensional torus, acting on the vector space assigned to the left vertex, acts on $i_{1}$ and $y_{s}$ with $j_{1}$ with opposite weights. We look at the space $\mathbb{C}^{2\ell -1}\setminus \{0\}$, formed by $y_{s}$’s and $j_{1}$. The $\mathbb{C}^{*}$-action on the one-dimensional framing attached to the right vertex allows to assume $j_{2}=1$. Observing that the action of the remaining $\mathbb{C}^{*}$ simultaneously rescales the vectors in $\mathbb{C}^{2\ell -1}\setminus \{0\}$, we recover the first $\mathbb{CP}^{2\ell -2}$ component in $\rho _s^{-1}(0)$. Similarly, if all $y_{s}=0$, one comes up with $\mathbb{CP}^{2\ell -2}$ with coordinates $x_{k}$ and $j_{2}$. It remains to notice that the projective spaces have exactly one point of intersection $(x_{s}=y_{k}=i_{1}=i_{2}=0, j_{1}=j_{2}=1)$.

Next we verify the assertion of (b). The moment map equations considered separately for the two vertices are equivalent to

$$\begin{equation} \begin{cases} \sum \limits _{i=1}^{\ell -1} (x_{i}y_{\ell +i} + x_{\ell +i}y_{i})+j_{1}i_{1}=0, \\ j_{1}i_{1} = j_{2}i_{2}. \end{cases} {\tag{10}} \end{equation}$$

Recall that $\theta = (-1,-1)$. Then the $\theta$-semistable locus consists of all representations for which at least one of $j_{1}, j_{2}$ is not equal to zero and

•

if $j_{1}\neq 0$ and $j_2=0$ there exists an $i$ such that $x_{i}\neq 0$;

•

if $j_{2}\neq 0$ and $j_1=0$ there exists a $j$ such that $y_{j}\neq 0$.

The formulas for the torus action below are derived from the fact that $x_{i}\in \operatorname {Hom}(r_{1},r_{2})$ and $y_{i}\in \operatorname {Hom}(r_{2},r_{1})$ are the elements above and below diagonal in the $i$th matrix of our quiver variety, where $r=r_{0}\oplus r_{1}\oplus r_{2}$ is the decomposition of the representation into simples.

$$\begin{equation} \begin{cases} t'_{1}x_{1}=t_1 x_{1}t_2^{-1}, \\ \mathinner{\ldotp \ldotp \ldotp }\\ t'^{-1}_{\ell -1}x_{2\ell -1}=t_1 x_{2\ell -1}t_2^{-1}, \\ t'_{1}y_{1}=t_1^{-1} y_{1}t_2, \\ \mathinner{\ldotp \ldotp \ldotp }\\ t'^{-1}_{\ell -1}y_{2\ell -1}=t_1^{-1} y_{2\ell -1}t_2, \\ j_{1}=t_1^{-1}j_{1}, \\ j_{2}=t_2^{-1}j_{2}, \\ i_{1}=t_1 i_{1}, \\ i_{2}=t_2 i_{2}, \end{cases} {\tag{11}} \end{equation}$$

here $(t'_1,\mathinner{\ldotp \ldotp \ldotp },t'_{\ell -1})\in T'$ and $(t_1,t_2)\in (\mathbb{C}^*)^2$. We first notice that it is not possible for both $i_{s}$ and $j_{s}$ to be nonzero ($s\in \{1,2\}$), as otherwise the second equation of 10 would imply all $i_{s}$, $j_{s}$ ($s\in \{1,2\}$) were nonzero and consequently $t_1=t_2=1$, implying all $x_{k}=y_{c}=0$, hence, contradicting the first equation of 10. It follows from (a) that $i_{1}=i_{2}=0$. From the system of equalities 11 it also follows that we must have one of the following

•

$x_{i}\neq 0, y_{l+i}\neq 0$ and $y_{i}\neq 0$ with $i \in \{2,\mathinner{\ldotp \ldotp \ldotp },\ell \}$;

•

$x_{l+i}\neq 0$ and $y_{i}\neq 0$ with $i \in \{2,\mathinner{\ldotp \ldotp \ldotp },\ell \}$;

•

all $x_{i}$ and all $y_{j}$ are zero with $j_{1}=j_{2}=1$ and $i_{1}=i_{2}=0$.

In each of the former two cases 10 reduces to either $x_{i}y_{\ell +i}= 0$ or $x_{\ell +i}y_{i} = 0$, then the claim of the proposition easily follows from the description of semistable points.

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Remark 3.4.

The slice $\mathcal{SL}_{p}\subset \overline{\mathcal{M}}^{\theta }(2,\ell )$ is a formal subscheme (formal neighborhood of the point $p$). We describe the intersection of the fixed point loci $\mathcal{SL}^{T'}_{p}\cap \overline{\mathcal{M}}^{\theta }(2,\ell )^{T'}$ (the latter was found in Remark 2.18). Each fixed point $(x_{s}=1, x_{\neq s}=y_{j}=i_{1}=i_{2}=j_{2}=0, j_{1}=1)$ on the slice with $s \in \{1,\mathinner{\ldotp \ldotp \ldotp },\ell -1\}$ is the fixed point $(X_{\ell }=\begin{pmatrix} 1 & 0 \\ 0 & -1\end{pmatrix}, Y_{\ell }=0$, $X_{s} = \begin{pmatrix} 0 & 1 \\ 0 & 0\end{pmatrix}, X_{\neq s}=Y_k=0, i=0, j=\begin{pmatrix} 1 \\ 0\end{pmatrix})$ on $\overline{\mathcal{M}}^{\theta }(2,\ell )^{T'}$; $(y_{s}=1, x_{i}=y_{\neq s}=i_{1}=i_{2}=j_{1}=0, j_{2}=1)$ is $(X_{\ell }=\begin{pmatrix} -1 & 0 \\ 0 & 1\end{pmatrix}, Y_{\ell }=0$, $X_{s} = \begin{pmatrix} 0 & 1 \\ 0 & 0\end{pmatrix}, X_{\neq s}=Y_k=0, i=0, j=\begin{pmatrix} 1 \\ 0\end{pmatrix})$. Notice that these points are respectively the points $(1,0)$ and $(-1,0)$ on $\mathbb{C}^2_s\subset \overline{\mathcal{M}}^{\theta }(2,\ell )^{T'}$ (see Remark 2.18). In case $s \in \{\ell +1,\mathinner{\ldotp \ldotp \ldotp },2\ell \}$ the fixed points on the slice are $(X_{\ell }=\begin{pmatrix} 1 & 0 \\ 0 & -1\end{pmatrix}, Y_{\ell }=0$, $Y_{s}=\begin{pmatrix} 0 & 1 \\ 0 & 0\end{pmatrix}, X_{\neq s}=Y_k=0, i=0, j=\begin{pmatrix} 1 \\ 0\end{pmatrix})$ and $(X_{\ell }=\begin{pmatrix} -1 & 0 \\ 0 & 1\end{pmatrix}, Y_{\ell }=0$, $Y_{s} = \begin{pmatrix} 0 & 1 \\ 0 & 0\end{pmatrix}, X_{\neq s}=Y_k=0, i=0, j=\begin{pmatrix} 1 \\ 0\end{pmatrix})$, finally, $(x_{s}=y_{k}=i_{1}=i_{2}=0, j_{1}=j_{2}=1)$ becomes $(X_{\ell }=\begin{pmatrix} 1 & 0 \\ 0 & -1\end{pmatrix},X_{\neq \ell }=Y_{k}=0, i=0, j=\begin{pmatrix} 1 \\ 1\end{pmatrix}) \in T^{*}\mathbb{P}^{1}$.

4. Category $\mathcal{O}_{\xi }(\mathcal{\overline{S}}_{\lambda }(2,\ell ))$ for the slice $\mathcal{SL}_{p}$

The main goal of this section is to provide a description of the category $\mathcal{O}_{\nu }(\mathcal{\overline{SL}}_{\lambda }(2,\ell ))$ for the slice $\mathcal{SL}_{p}$. These results will be used in the next section for the study of category $\mathcal{O}_{\nu }(\mathcal{\overline{A}}_{\lambda }(2,\ell ))$. As $\mathcal{SL}_{p}$ is a hypertoric variety, we use the results of 6 and 7, where analogous categories were explicitly described in a more general setting.

We start by briefly recalling the basic definitions, notions and results (for a more detailed exposition see 6 and 7).

4.1. Hypertoric varieties (a brief overview)

Let $\mathbb{D}$ be the Weyl algebra of polynomial differential operators on $\mathbb{C}^n$, i.e.

$$\begin{equation*} \mathbb{D} =\mathbb{C}\langle x_1,\partial _1,\mathinner{\ldotp \ldotp \ldotp }x_n,\partial _n\rangle , \end{equation*}$$

with $[x_i,x_j]=[\partial _i,\partial _j]=0$ and $[\partial _i,x_j]=\delta _{ij}$. The action of torus $\widetilde{T}=(\mathbb{C}^*)^n$ on $\mathbb{C}^n$ induces an action on $\mathbb{D}$. This provides the $\mathbb{Z}^n$-grading

$$\begin{equation*} \mathbb{D}=\underset{z\in W_\mathbb{Z}}{\bigoplus }\mathbb{D}_z, \end{equation*}$$

where $W_\mathbb{Z}$ is the character lattice of $\widetilde{T}$, deg$(x_i)=-\operatorname {deg}(\partial _i)=(0,\mathinner{\ldotp \ldotp \ldotp },0,\underset{i}{1},0,\mathinner{\ldotp \ldotp \ldotp },0)$ and $\mathbb{D}_z\coloneq \{a\in \mathbb{D}\nobreakspace |\nobreakspace t\cdot a=t_1^{z_1}\mathinner{\ldotp \ldotp \ldotp }t_n^{z_n}a\nobreakspace \forall\nobreakspace t\in \widetilde{T}\}$.

Observe that the $0^{\mathrm{th}}$ graded piece is $\mathbb{D}^{\widetilde{T}}=\mathbb{C}[x_1\partial _1,\mathinner{\ldotp \ldotp \ldotp }, x_n\partial _n]$ and define $h_i^-\coloneq \partial _ix_i$ and $h_i^+\coloneq x_i\partial _i$ with $h_i^--h_i^+=1$. We consider the Bernstein filtration on $\mathbb{D}$ (here deg$(x_i)=$ deg$(\partial _j)=1$) and let $H\coloneq \operatorname {gr}(\mathbb{D}_0)=\mathbb{C}[h_1,\mathinner{\ldotp \ldotp \ldotp },h_n]$, where $h_i\coloneq h_i^++F_0(\mathbb{D}_0)=h_i^-+F_0(\mathbb{D}_0)$.

Fix a direct summand $\Lambda _0 \subset W_\mathbb{Z}$, let $W_{\mathbb{R}}\coloneq W_{\mathbb{Z}}\otimes _{\mathbb{Z}}\mathbb{R}, V_{0,\mathbb{R}}=\mathbb{R}\Lambda _0, V_0\coloneq \mathbb{C}\Lambda _0\subset W\cong t^*$, $\mathfrak{k}=V_0^{\perp }$ and $K \subset \widetilde{T}$ be the connected subtorus with Lie algebra $\mathfrak{k}$. Thus $\Lambda _0$ may be identified with the character lattice of $\widetilde{T}/K$ and $W_\mathbb{Z}/\Lambda _0$ may be identified with the character lattice of $K$.

Consider the moment map for the action of torus $K\subset \tilde{T}=(\mathbb{C}^*)^n$ on $T^*\mathbb{C}^n$, i.e.

$$\begin{equation*} \mu : T^*\mathbb{C}^n \rightarrow \mathfrak{k}^*. \end{equation*}$$

Definition 4.1.

The hypertoric variety associated to the pair $(\Lambda _0, \eta )$ with $\eta$ a $\Lambda _0$-orbit in $W_{\mathbb{Z}}$ is $\mathfrak{M}(X)\coloneq \mu ^{-1}(0)^{\eta -ss}//K$. Also define $\mathfrak{M}_0(X)\coloneq \mu ^{-1}(0)//K$. We consider the categorical quotient in both cases. The projective map $\mathfrak{M}(X)\rightarrow \mathfrak{M}_0(X)$ will be denoted by $\kappa$. We will denote the subspace $\eta +V_{0,\mathbb{R}}\subset W_{\mathbb{R}}$ by $V_{\eta }$.

Let $\mathbb{S} \coloneq \mathbb{C}^*$ act on $T^*\mathbb{C}^n$ by inverse scalar multiplication, i.e. $s \cdot (z, w) \coloneq (s^{-1}z, s^{-1}w)$. This induces an $\mathbb{S}$-action on both $\mathfrak{M}(X)$ and $\mathfrak{M}_0(X)$, and the map $\kappa$ is $\mathbb{S}$-equivariant. We have that $\kappa :\mathfrak{M}(X)\rightarrow \mathfrak{M}_0(X)$ is a conical symplectic resolution. The symplectic form $\omega$ has weight $2$ w.r.t. the aforementioned $\mathbb{S}$-action.

Definition 4.2.

The hypertoric enveloping algebra associated to $\Lambda _0$ is the ring of $K$-invariants $U \coloneq \mathbb{D}^K =\underset{z\in \Lambda _0}{\bigoplus }\mathbb{D}_z$.

Remark 4.3.

The hypertoric variety $\mathfrak{M}_0(X)$ is affine, and for any central character $\lambda$ of the hypertoric enveloping algebra $U$ there is a natural isomorphism $\operatorname {gr}(U_\lambda ) \simeq \mathbb{C}[\mathfrak{M}_0] \simeq \mathbb{C}[\mathfrak{M}]$ (Proposition $5.2$ in 7).

4.2. Hypertoric category $\mathcal{O}$

Let $Z(U)$ denote the center of $U$. It is not hard to show that $Z(U)$ is the subalgebra isomorphic to the image of $S[\mathfrak{k}]$ under the quantum comoment map (Section $3.2$ of 7). Let $\lambda : Z(U)\rightarrow \mathbb{C}$ be a central character. Notice that the isomorphism $Z(U)\simeq S[\mathfrak{k}]$ allows to think of $\lambda$ as an element of $\mathfrak{k}^*$. We will denote by $U_\lambda \coloneq U/\langle ker(\lambda )\rangle U$ the corresponding central quotient. Consider a module $M\in U\operatorname {-mod}$. For a point $v\in \mathbf{W}\coloneq \operatorname {Specm}(\mathbb{C}[h_1^\pm ,\mathinner{\ldotp \ldotp \ldotp }, h_n^\pm ]/\langle h_i^+-h_i^-+1\nobreakspace |\nobreakspace i\in 1,\mathinner{\ldotp \ldotp \ldotp }, n\rangle )$, let $\mathcal{J}_v\subset \mathbb{C}[h_1^\pm ,\mathinner{\ldotp \ldotp \ldotp }, h_n^\pm ]/\langle h_i^+-h_i^-+1\nobreakspace |\nobreakspace i\in 1,\mathinner{\ldotp \ldotp \ldotp }, n\rangle$ denote the corresponding maximal ideal. Then the generalized $v$-weight space of $M$ is defined as

$$\begin{equation*} M_v\coloneq \{m\in M\nobreakspace |\nobreakspace \mathcal{J}^k_v m=0 \text{ for } k\gg 0 \}. \end{equation*}$$

The support of $M$ is defined by

$$\begin{equation*} \operatorname {Supp}M\coloneq \{v \in \mathbf{W}\nobreakspace |\nobreakspace M_v\neq 0 \}. \end{equation*}$$

We will use the notation $U\operatorname {-mod}_{\Lambda }$ for $M \in U\operatorname {-mod}$ with $\operatorname {Supp}M\subset \Lambda$.

Choose a generic element $\xi \in \Lambda _0^*\simeq (t/\mathfrak{k})^*$, the action of $\xi$ lifts to $U$ and produces a grading given by

$$\begin{equation*} U^k\coloneq \underset{\xi (z)=k}{\bigoplus }U_z. \end{equation*}$$

Set

$$\begin{equation*} U^+\coloneq \underset{k\geq 0}{\bigoplus }U^k \text{ and } U^-\coloneq \underset{k\leq 0}{\bigoplus }U^k, \end{equation*}$$

similarly, $U^+_\lambda$ and $U^-_\lambda$ are the images of $U^+$ and $U^-$ under the quotient map $U \rightarrow U_\lambda$.

Definition 4.4.

The hypertoric category $\mathcal{O}$ is the full subcategory of $U$-mod consisting of modules that are $U^+$-locally finite and semisimple over the center $Z(U)$. Define $\mathcal{O}_{\lambda }$ to be the full subcategory of $\mathcal{O}$ consisting of modules on which $U$ acts with central character $\lambda$. Equivalently, it is as the full subcategory of $U_\lambda$-mod consisting of modules that are $U^+_\lambda$-locally finite. Finally, define $\mathcal{O}(\Lambda _0, \Lambda , \xi )$ to be the full subcategory of $\mathcal{O}_{\lambda }$ consisting of modules supported in $\Lambda$; equivalently, the full subcategory of $U_\lambda \operatorname {-mod}_{\Lambda }$ consisting of modules that are $U^+_\lambda$-locally finite. The triple $\mathbf{X}\coloneq (\Lambda _0, \Lambda , \xi )$ is called a quantized polarized arrangement.

Similarly to category $\mathcal{O}$ of a semisimple Lie algebra, we have the direct sum decompositions

$$\begin{equation} \begin{aligned} \mathcal{O} = \underset{\Lambda \in W/\Lambda _0}{\bigoplus } \mathcal{O}(\Lambda _0, \Lambda , \xi ) \text{ and } \\ \mathcal{O}_{\lambda } = \underset{\Lambda ' \in V_{\lambda }/\Lambda _0}{\bigoplus } \mathcal{O}(\Lambda _0, \Lambda ', \xi ). \end{aligned} {\tag{12}} \end{equation}$$

The summands in the decompositions above are blocks, i.e. they are the smallest possible direct summands (see Section $4.1$ of 7 for details).

Set $V_{\lambda } \coloneq \lambda + V_0 = \lambda + \mathbb{C}\Lambda _0, V_{\lambda ,\mathbb{R}} \coloneq \lambda + \mathbb{R}\Lambda _0$ and let $\mathbf{\Lambda }$ be a $\Lambda _0$-orbit in $W$ with $I_\Lambda$ the set of indices $i\in \{1,\mathinner{\ldotp \ldotp \ldotp },n\}$ for which $h^+_i(\Lambda )\subset \mathbb{Z}$ (equivalently $h^-_i(\Lambda )\subset \mathbb{Z}$. For a sign vector $\alpha \in \{+,-\}^{n}$ define the chamber $P_{\alpha ,0}$ to be the subset of the affine space $V_{\Lambda }\coloneq \Lambda + V_{0,\mathbb{R}}\subset W_{\mathbb{R}}$ cut out by the inequalities

$$\begin{equation*} h_i \geq 0 \text{ for all } i \in I_{\mathbf{\Lambda }} \text{ with } \alpha (i) = + \text{ and } h_i \leq 0 \text{ for all } i \in I_{\mathbf{\Lambda }} \text{ with } \alpha (i) = -. \end{equation*}$$

If $P_\alpha \cap \Lambda$ is nonempty, we say that $\alpha$ is feasible for $\Lambda$. We call $\alpha$ bounded for $\xi$ if the restriction of $\xi$ is proper and bounded above on $P_\alpha$. The set of feasible sign vectors will be denoted by $\mathcal{F}_{\Lambda }$, the set of bounded vectors by $\mathcal{B}_{\xi }$ and the set of bounded feasible vectors by $\mathcal{P}_{\Lambda ,\xi }\coloneq \mathcal{F}_{\Lambda }\cap \mathcal{B}_{\xi }$.

Example 4.5.

In case $\ell =2$, the slice $\mathcal{SL}_{p}$ is the hypertoric variety obtained from the $K=(\mathbb{C}^*)^2$-action on $\mathbb{C}^4$ via

$$\begin{equation*} t\cdot (x_{1}, x_{2}, i_{1}, i_{2})=(t_{1}^{-1}t_{2}x_1,t_{1}t_{2}^{-1}x_2,t_{1}^{-1}i_1,t_{2}^{-1}i_2). \end{equation*}$$

Notice that $\mathfrak{k}\hookrightarrow Lie(\widetilde{T})$ and the image is span$((-1,1,-1,0),(1,-1,0,-1))$, set $L\!\coloneq \! \operatorname {span}_{\mathbb{R}}((-1,1,-1,0),(1,-1,0,-1))$. Then $V_{0,\mathbb{R}}\!=\!\operatorname {span}_{\mathbb{R}}((1,1,0,0),(0,1,1,-1))$ (the subspace of $W_{\mathbb{R}}$ orthogonal to $L$) and we consider $\Lambda _0=V_{0,\mathbb{R}}\cap W_{\mathbb{Z}}$ and the central character $\lambda : S[\mathfrak{k}]\rightarrow \mathbb{C}$ defined by $\lambda (t_1,t_2)=(\tilde{\lambda },\tilde{\lambda })$ for $\tilde{\lambda } \in \mathbb{C}$. We take $\eta =(-1,-1)$ to be the restriction of the character $\theta$ of $G$.

Then $V_{\lambda }$ is cut out in $W$ (or $V_{\lambda ,\mathbb{R}}$ inside $W_{\mathbb{R}}$) by the following equations:

$$\begin{equation*} \begin{cases} -x_{1}+x_{2}-i_{1}=\tilde{\lambda } \\ \quad x_{1}-x_{2}-i_{2}=\tilde{\lambda } \end{cases}, \end{equation*}$$ equivalently,

$$\begin{equation*} \begin{cases} \; i_{1}+i_{2}=-2\tilde{\lambda } \\ x_{1}-x_{2}=i_{2}+\tilde{\lambda } \end{cases}. \end{equation*}$$

This is a $2$-dimensional affine subspace of $W$. We identify $V_{\lambda }$ with $\mathbb{C}^2$ (or $V_{\lambda ,\mathbb{R}}$ with $\mathbb{R}^2$) by choosing the origin of $V_{\lambda }$ to be the point $(0,0,-\tilde{\lambda },-\tilde{\lambda })$ and the basis $u_1\coloneq (1,1,0,0), u_2\coloneq (0,1,1,-1)$. Next we pick a one-parameter subgroup $\xi =(2,1)$. In case $\tilde{\lambda } \in \mathbb{Z}$, we have $\mathcal{P}_{\lambda ,\xi }=\{+---,-+--,----,---+,--+-\}$ (see Figures 2 and 3 for a depiction of the corresponding polarized arrangement, chambers and sign vectors).

Remark 4.6.

If $\alpha \in \mathcal{P}_{\Lambda ,\xi }$ and $\xi$ is a generic character then the differential of $\xi$ attains its maximal value at a single point of $P_\alpha$. This point will be denoted by $a_{\alpha }$. It is the intersection of $\operatorname {dim}(V_{\lambda })$ hyperplanes from

$$\begin{equation*} h^+_i = 0 \text{ for all } i \in I_{\mathbf{\Lambda }} \text{ with } \alpha (i) = + \text{ and } h^-_i = 0 \text{ for all } i \in I_{\mathbf{\Lambda }} \text{ with } \alpha (i) = -. \end{equation*}$$

Let $C_\alpha$ be the unique polyhedral cone cut out in $V_{\lambda }$ by $\operatorname {dim}V_{\lambda }$ inequalities

$$\begin{equation*} h^+_i \geq 0 \text{ for all } i \in I_{\mathbf{\Lambda }}, h^+_i(a_{\alpha })=0 \text{ with } \alpha (i) = + \text{ and } \end{equation*}$$

$$\begin{equation*} h^-_i \leq 0 \text{ for all } i \in I_{\mathbf{\Lambda }}, h^-_i(a_{\alpha })=0 \text{ with } \alpha (i) = -. \end{equation*}$$ Notice that $P_\alpha \subset C_\alpha$ and the differential of $\xi$ is negative on the extremal rays of $C_\alpha$.

Next we describe the standard objects of $\mathcal{O}(\mathbf{X})$. For any sign vector $\alpha \in \{+,-\}^{I_{\mathbf{\Lambda }}}$, consider the $\mathbb{D}$-module

$$\begin{equation*} \triangle _\alpha \coloneq \mathbb{D}/I_{\alpha }, \end{equation*}$$

where $I_{\alpha }$ is the left ideal generated by the elements

•

$\partial _i,\text{ for all } i \text{, s.t. } h_i^+(a_{\alpha })=0$,

•

$x_i,\text{ for all } i \text{, s.t. } h_i^-(a_{\alpha })=0$,

•

$h^+_i -h^+_i (a_{\alpha }), i\not \in I_{\mathbf{\Lambda }}$.

Define $\triangle _\alpha ^\Lambda \coloneq \underset{v\in \mathbb{\Lambda }}{\bigoplus } (\triangle _\alpha )_v$, then the standard objects of $\mathcal{O}(\Lambda _0, \Lambda , \xi )$ are $\triangle _\alpha ^\Lambda$ for $\alpha \in \mathcal{P}_{\Lambda ,\xi }$ (see Section $4.4$ in 7). Let $S_\alpha ^\Lambda$ denote the unique simple quotient of $\triangle _\alpha ^\Lambda$.

We will need one more definition.

Definition 4.7.

The quantized polarized arrangement $\mathbf{X}=(\Lambda _0,\Lambda ,\xi )$ and polarized arrangement $X=(\Lambda _0,\eta ,\xi )$ are said to be linked if $\pi (\mathcal{F}_{\Lambda })=\mathcal{F}_{\eta }$ for the projection $\pi : \{+,-\}^n\rightarrow \{+,-\}^{I_{\Lambda }}$.

Remark 4.8.

The hypertoric category $\mathcal{O}_{\lambda }$ is a category $\mathcal{O}_{\xi }$ for $U_\lambda$ in the sense of Definition 1.5.

Remark 4.9.

If $\mathbf{X}=(\Lambda _0, \Lambda , \xi )$ is regular, then the category $\mathcal{O}(\mathbf{X})$ is highest weight and Koszul (see Definition $2.10$ and Corollary $4.10$ in 7).

4.3. Hypertoric category $\mathcal{O}$ for the slice $\mathcal{SL}_{p}$

The slice $\mathcal{SL}_{p}$ is the hypertoric variety obtained from the $K=(\mathbb{C}^*)^2$-action on $\mathbb{C}^{2\ell }$ via

$$\begin{align*} & (t_1,t_2)\cdot (x_{1}, \mathinner{\ldotp \ldotp \ldotp }, x_{\ell -1}, x_{\ell }, \mathinner{\ldotp \ldotp \ldotp }, x_{2\ell -2}, i_{1}, i_{2}) \\ & =(t_{1}^{-1}t_{2}x_1,\mathinner{\ldotp \ldotp \ldotp }, t_{1}^{-1}t_{2}x_{\ell -1}, t_{1}t_{2}^{-1}x_{\ell },\mathinner{\ldotp \ldotp \ldotp },t_{1}t_{2}^{-1}x_{2\ell -2},t_{1}^{-1}i_1,t_{2}^{-1}i_2), \end{align*}$$

it can be also viewed as a quiver variety (see Figure 1). This is the toric variety $\mathfrak{M}(X)$ for the polarized arrangement $X=(\Lambda _0, \eta ,\xi )$. Let $\Lambda _0=V_{0,\mathbb{R}}\cap W_{\mathbb{Z}}$ for $V_{0,\mathbb{R}}=\operatorname {span}_{\mathbb{R}}(u_1,\mathinner{\ldotp \ldotp \ldotp },u_{2\ell -2})$ (where the vectors $u_1$, …, $u_{2\ell -2}$ are defined below) and the same character $\lambda$ and same $\eta$ as for $\ell =2$ above, then $V_{\lambda }$ is the affine subset of $W$ given by

$$\begin{equation*} \begin{cases} -\sum \limits _{k=1}^{\ell -1}x_{k}+\sum \limits _{k=1}^{\ell -1}x_{\ell -1+k}-i_{1}=\tilde{\lambda } \\ \quad \sum \limits _{k=1}^{\ell -1}x_{k}-\sum \limits _{k=1}^{\ell -1}x_{\ell -1+k}-i_{2}=\tilde{\lambda } \end{cases}, \end{equation*}$$ equivalently,

$$\begin{equation*} \begin{cases} \; i_{1}+i_{2}=-2\tilde{\lambda } \\ \sum \limits _{k=1}^{\ell -1}x_{k}-\sum \limits _{k=1}^{\ell -1}x_{\ell -1+k}=i_{2}+\tilde{\lambda } \end{cases}. \end{equation*}$$

This is a $2\ell -2$-dimensional affine subspace of $W$. We choose the origin to be the point $(0,\mathinner{\ldotp \ldotp \ldotp },0,0,-\tilde{\lambda },-\tilde{\lambda })$ and the basis

$$\begin{flalign*} & u_1\coloneq (1,-1,0,\mathinner{\ldotp \ldotp \ldotp },0,0), \\ & u_2\coloneq (1,0,-1,\mathinner{\ldotp \ldotp \ldotp },0,0), \\ &\nobreakspace\nobreakspace\nobreakspace\nobreakspace\nobreakspace\nobreakspace\nobreakspace\nobreakspace\nobreakspace\nobreakspace\nobreakspace \mathinner{\ldotp \ldotp \ldotp }\\ & u_{\ell -1}\coloneq (1,0,\mathinner{\ldotp \ldotp \ldotp },-1,0,\mathinner{\ldotp \ldotp \ldotp },0,0), \\ & u_{\ell }\coloneq (1,0,\mathinner{\ldotp \ldotp \ldotp },0,1,0,\mathinner{\ldotp \ldotp \ldotp },0,0), \\ & u_{\ell +1}\coloneq (1,0,\mathinner{\ldotp \ldotp \ldotp },0,0,1,0,\mathinner{\ldotp \ldotp \ldotp },0,0), \\ & u_{2\ell -3}\coloneq (1,0,\mathinner{\ldotp \ldotp \ldotp },0,1,0,0), \\ & u_{2\ell -2}\coloneq (0,\mathinner{\ldotp \ldotp \ldotp },0,1,1,-1). \end{flalign*}$$

One convenient choice of a character is $\xi =(1,\mathinner{\ldotp \ldotp \ldotp },\ell -2,\ell ,\mathinner{\ldotp \ldotp \ldotp },2(\ell -1),\ell -1)$.

Definition 4.10.

The algebra $\mathcal{\overline{S}}_{\lambda }(2,\ell )$ will stand for the quantization of the slice $\mathcal{SL}_{p}$ with period