2.1 Conical symplectic resolutions and their deformations

Let $X_{0}$ be a normal Poisson affine variety equipped with an action of $\mathbb{C}^{\times }$ such that the grading on $\mathbb{C}[X_{0}]$ is positive (meaning that the graded component $\mathbb{C}[X_{0}]_{i}$ is zero when $i<0$ , and $\mathbb{C}[X_{0}]_{0}=\mathbb{C}$ ) and there is a positive integer $d$ such that $\{\mathbb{C}[X_{0}]_{i},\mathbb{C}[X_{0}]_{j}\}\subset \mathbb{C}[X_{0}]_{i+j-d}$ . By a symplectic resolution of singularities of $X_{0}$ , we mean a pair $(X,\unicode[STIX]{x1D70C})$ of:

• ∙ a symplectic algebraic variety $X$ (with form $\unicode[STIX]{x1D714}$ );

• ∙ a morphism $\unicode[STIX]{x1D70C}$ of Poisson varieties that is a projective resolution of singularities.

Below we assume that $(X,\unicode[STIX]{x1D70C})$ is a symplectic resolution of singularities. Besides, we will assume that $(X,\unicode[STIX]{x1D70C})$ is conical meaning that the $\mathbb{C}^{\times }$ -action lifts to $X$ in such a way that $\unicode[STIX]{x1D70C}$ is equivariant. This $\mathbb{C}^{\times }$ -action will be called contracting later on.

Note that $\unicode[STIX]{x1D70C}^{\ast }:\mathbb{C}[X_{0}]\rightarrow \mathbb{C}[X]$ is an isomorphism because $X_{0}$ is normal. By the Grauert–Riemenschneider theorem, we have $H^{i}(X,{\mathcal{O}}_{X})=0$ for $i>0$ . By results of Kaledin, [Reference KaledinKal06b, Theorem 2.3], $X_{0}$ has finitely many symplectic leaves.

We will be interested in deformations $\hat{X}/\mathfrak{p}$ , where $\mathfrak{p}$ is a finite-dimensional vector space, and $\hat{X}$ is a symplectic scheme over $\mathfrak{p}$ with symplectic form $\hat{\unicode[STIX]{x1D714}}\in \unicode[STIX]{x1D6FA}^{2}(\hat{X}/\mathfrak{p})$ that specializes to $\unicode[STIX]{x1D714}$ and also with a $\mathbb{C}^{\times }$ -action on $\hat{X}$ having the following properties:

• ∙ the morphism $\hat{X}\rightarrow \mathfrak{p}$ is $\mathbb{C}^{\times }$ -equivariant;

• ∙ the restriction of the action to $X$ coincides with the contracting action;

• ∙ $t.\hat{\unicode[STIX]{x1D714}}:=t^{d}\hat{\unicode[STIX]{x1D714}}$ .

It turns out that there is a universal such deformation $\tilde{X}$ over $\tilde{\mathfrak{p}}:=H^{2}(X,\mathbb{C})$ (any other deformation is obtained via the pull-back with respect to a linear map $\mathfrak{p}\rightarrow \tilde{\mathfrak{p}}$ ). Moreover, the deformation $\tilde{X}$ is trivial in the category of $C^{\infty }$ -manifolds. This result is due to Namikawa [Reference NamikawaNam08, Lemmas 12 and 22 and Proposition 13].

Now we are going to review some structural features of conical symplectic resolutions mostly due to Kaledin and Namikawa.

First of all, let us point out that $X$ has no odd cohomology [Reference Braden, Proudfoot and WebsterBPW16, Proposition 2.5].

For $\unicode[STIX]{x1D706}\in \tilde{\mathfrak{p}}$ , let us write $X_{\unicode[STIX]{x1D706}}$ for the corresponding fiber of $\tilde{X}\rightarrow \tilde{\mathfrak{p}}$ . It turns out that, for $\unicode[STIX]{x1D706}$ Zariski generic, $X_{\unicode[STIX]{x1D706}}$ is affine and independent of the choice of a conical symplectic resolution $X$ . This shows that the groups $H^{i}(X,\mathbb{Z})$ are independent of the choice of $X$ . Furthermore, if $X,X^{\prime }$ are two conical symplectic resolutions of $X$ , then there are open subvarieties $X^{0}\subset X,{X^{\prime }}^{0}\subset X^{\prime }$ with complements of codimension bigger than $1$ that are isomorphic; see, e.g., [Reference Braden, Proudfoot and WebsterBPW16, Proposition 2.18]. This allows us to identify the Picard groups $\operatorname{Pic}(X)=\operatorname{Pic}(X^{\prime })$ . Moreover, the Chern class map defines an isomorphism $\mathbb{C}\,\otimes _{\mathbb{Z}}\operatorname{Pic}(X)\xrightarrow[{}]{{\sim}}H^{2}(X,\mathbb{C})$ . The isomorphisms $\mathbb{C}\,\otimes _{\mathbb{Z}}\operatorname{Pic}(X)\xrightarrow[{}]{{\sim}}H^{2}(X,\mathbb{C})$ , $\mathbb{C}\,\otimes _{\mathbb{Z}}\operatorname{Pic}(X^{\prime })\xrightarrow[{}]{{\sim}}H^{2}(X^{\prime },\mathbb{C})$ intertwine the identifications $\operatorname{Pic}(X)\cong \operatorname{Pic}(X^{\prime })$ , $H^{2}(X,\mathbb{C})\cong H^{2}(X^{\prime },\mathbb{C})$ . Let $\tilde{\mathfrak{p}}_{\mathbb{Z}}$ be the image of $\operatorname{Pic}(X)$ in $H^{2}(X,\mathbb{C})$ .

Set $\tilde{\mathfrak{p}}_{\mathbb{R}}:=\mathbb{R}\otimes _{\mathbb{Z}}\tilde{\mathfrak{p}}_{\mathbb{Z}}$ . There is a finite group $W$ acting on $\tilde{\mathfrak{p}}_{\mathbb{R}}$ as a reflection group, such that the movable cone $C$ of $X$ (that does not depend on the choice of a resolution) is a fundamental chamber for $W$ . We can partition $C$ into the union of cones $C_{1},\ldots ,C_{m}$ (with disjoint interiors) such that the possible conical symplectic resolutions of $X_{0}$ are in one-to-one correspondence with $C_{1},\ldots ,C_{m}$ in such a way that the cone corresponding to a resolution $X^{\prime }$ is its ample cone. Let $H_{1},\ldots ,H_{k}$ be the hyperplanes spanned by the codimension- $1$ faces of the cones $C_{1},\ldots ,C_{m}$ . Set

(2.1) $$\begin{eqnarray}{\mathcal{H}}_{\mathbb{C}}:=\mathop{\bigcup }_{1\leqslant i\leqslant k,w\in W}w(\mathbb{C}\otimes _{\mathbb{R}}H_{i}).\end{eqnarray}$$

Then $X_{\unicode[STIX]{x1D706}}$ is affine if and only if $\unicode[STIX]{x1D706}\not \in {\mathcal{H}}_{\mathbb{C}}$ . The results in this paragraph are due to Namikawa [Reference NamikawaNam15].

For $\unicode[STIX]{x1D703}\not \in {\mathcal{H}}_{\mathbb{C}}$ , we will write $X^{\unicode[STIX]{x1D703}}$ for the resolution corresponding to the ample cone containing $W\unicode[STIX]{x1D703}\,\cap \,C$ . Further, if $w\unicode[STIX]{x1D703}\in C$ , we will choose a different identification of $H^{2}(X^{\unicode[STIX]{x1D703}},\mathbb{C})$ with $\tilde{\mathfrak{p}}$ , one twisted by $w$ (so that the ample cone actually contains $\unicode[STIX]{x1D703}$ ).

2.2 Quantizations

We will study quantizations of $X,X_{0},\tilde{X}$ . By a quantization of $X_{0}$ , we mean a filtered algebra ${\mathcal{A}}$ together with an isomorphism $\operatorname{gr}{\mathcal{A}}\cong \mathbb{C}[X_{0}]$ of graded Poisson algebras. By a quantization of $X=X^{\unicode[STIX]{x1D703}}$ , we mean a sheaf ${\mathcal{A}}^{\unicode[STIX]{x1D703}}$ of filtered algebras in the conical topology on $X$ (in this topology, ‘open’ means Zariski open and $\mathbb{C}^{\times }$ -stable) that is complete and separated with respect to the filtration together with an isomorphism $\operatorname{gr}{\mathcal{A}}^{\unicode[STIX]{x1D703}}\cong {\mathcal{O}}_{X^{\unicode[STIX]{x1D703}}}$ (of sheaves of graded Poisson algebras). A quantization $\tilde{{\mathcal{A}}}^{\unicode[STIX]{x1D703}}$ of the universal deformation $\tilde{X}^{\unicode[STIX]{x1D703}}$ is:

• ∙ a filtered sheaf of $\mathbb{C}[\tilde{\mathfrak{p}}]$ -algebras (with a complete and separated filtration), where the induced filtration on $\mathbb{C}[\tilde{\mathfrak{p}}]$ coincides with the natural one, where $\deg \tilde{\mathfrak{p}}=d$ ;

• ∙ together with an isomorphism $\operatorname{gr}\tilde{{\mathcal{A}}}^{\unicode[STIX]{x1D703}}\xrightarrow[{}]{{\sim}}{\mathcal{O}}_{\tilde{X}^{\unicode[STIX]{x1D703}}}$ of graded Poisson $\mathbb{C}[\tilde{\mathfrak{p}}]$ -algebras.

Note that we can specialize a quantization $\tilde{{\mathcal{A}}}^{\unicode[STIX]{x1D703}}$ to a point in $\tilde{\mathfrak{p}}$ getting a quantization of $X$ .

A result of Bezrukavnikov and Kaledin [Reference Bezrukavnikov and KaledinBK04] (with ramifications given in [Reference LosevLos12, § 2]) shows that quantizations ${\mathcal{A}}^{\unicode[STIX]{x1D703}}$ are parameterized (up to an isomorphism) by the points in $\tilde{\mathfrak{p}}$ . More precisely, there is a canonical quantization $\tilde{{\mathcal{A}}}^{\unicode[STIX]{x1D703}}$ of $\tilde{X}^{\unicode[STIX]{x1D703}}$ such that the quantization of $X^{\unicode[STIX]{x1D703}}$ corresponding to $\unicode[STIX]{x1D706}\in \tilde{\mathfrak{p}}$ is the specialization of $\tilde{{\mathcal{A}}}^{\unicode[STIX]{x1D703}}$ to $\unicode[STIX]{x1D706}$ . The quantization $\tilde{{\mathcal{A}}}^{\unicode[STIX]{x1D703}}$ has the following important property: there is an anti-automorphism, $\unicode[STIX]{x1D70D}$ (to be called the parity anti-automorphism), that preserves the filtration, is the identity on the associated graded algebra, preserves $\tilde{\mathfrak{p}}^{\ast }\subset \unicode[STIX]{x1D6E4}(\tilde{{\mathcal{A}}}^{\unicode[STIX]{x1D703}})$ and induces $-1$ on $\tilde{\mathfrak{p}}^{\ast }$ . It follows that ${\mathcal{A}}_{-\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D703}}$ is isomorphic to $({\mathcal{A}}_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D703}})^{\text{opp}}$ .

We set $\tilde{{\mathcal{A}}}:=\unicode[STIX]{x1D6E4}(\tilde{{\mathcal{A}}}^{\unicode[STIX]{x1D703}}),{\mathcal{A}}_{\unicode[STIX]{x1D706}}=\unicode[STIX]{x1D6E4}({\mathcal{A}}_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D703}})$ , so that ${\mathcal{A}}_{\unicode[STIX]{x1D706}}$ is the specialization of $\tilde{{\mathcal{A}}}$ to $\unicode[STIX]{x1D706}$ . It follows from [Reference Braden, Proudfoot and WebsterBPW16, § 3.3] that the algebras $\tilde{{\mathcal{A}}},{\mathcal{A}}_{\unicode[STIX]{x1D706}}$ are independent of the choice of $\unicode[STIX]{x1D703}$ . From $H^{i}(X^{\unicode[STIX]{x1D703}},{\mathcal{O}}_{X^{\unicode[STIX]{x1D703}}})=0$ , we deduce that the higher cohomologies of both $\tilde{{\mathcal{A}}}^{\unicode[STIX]{x1D703}}$ and ${\mathcal{A}}_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D703}}$ vanish and also that $\tilde{{\mathcal{A}}}$ is a quantization of $\mathbb{C}[\tilde{X}]$ and ${\mathcal{A}}_{\unicode[STIX]{x1D706}}$ is a quantization of $\mathbb{C}[X]=\mathbb{C}[X_{0}]$ . Also, we note that ${\mathcal{A}}_{\unicode[STIX]{x1D706}}$ is the specialization of $\tilde{{\mathcal{A}}}$ at $\unicode[STIX]{x1D706}$ and that ${\mathcal{A}}_{-\unicode[STIX]{x1D706}}\cong {\mathcal{A}}_{\unicode[STIX]{x1D706}}^{\text{opp}}$ . Also, we have ${\mathcal{A}}_{\unicode[STIX]{x1D706}}\cong {\mathcal{A}}_{w\unicode[STIX]{x1D706}}$ for all $\unicode[STIX]{x1D706}\in \tilde{\mathfrak{p}},w\in W$ ; see [Reference Braden, Proudfoot and WebsterBPW16, § 3.3].

2.3 Localization and translation equivalences

Consider the categories $\operatorname{Coh}({\mathcal{A}}_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D703}})$ of coherent ${\mathcal{A}}_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D703}}$ -modules and ${\mathcal{A}}_{\unicode[STIX]{x1D706}}\text{-}\!\operatorname{mod}$ of finitely generated ${\mathcal{A}}_{\unicode[STIX]{x1D706}}$ -modules. Recall that an ${\mathcal{A}}_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D703}}$ -module $M$ is called coherent if it can be equipped with a complete and separated ${\mathcal{A}}_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D703}}$ -module filtration such that $\operatorname{gr}M$ is coherent (such a filtration is called good).

We have functors between these categories, the global section functor $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D706}}:\operatorname{Coh}({\mathcal{A}}_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D703}})\rightarrow {\mathcal{A}}_{\unicode[STIX]{x1D706}}\text{-}\!\operatorname{mod}$ and its left adjoint, the localization functor $\operatorname{Loc}_{\unicode[STIX]{x1D706}}:{\mathcal{A}}_{\unicode[STIX]{x1D706}}\text{-}\!\operatorname{mod}\rightarrow \operatorname{Coh}({\mathcal{A}}_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D703}}),N\mapsto {\mathcal{A}}_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D703}}\,\otimes _{{\mathcal{A}}_{\unicode[STIX]{x1D706}}}N$ . We can also consider their derived functors $R\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D706}}:D^{b}(\operatorname{Coh}({\mathcal{A}}_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D703}}))\rightarrow D^{b}({\mathcal{A}}_{\unicode[STIX]{x1D706}}\text{-}\!\operatorname{mod})$ and $L\operatorname{Loc}_{\unicode[STIX]{x1D706}}:D^{-}({\mathcal{A}}_{\unicode[STIX]{x1D706}}\text{-}\!\operatorname{mod})\rightarrow D^{-}(\operatorname{Coh}({\mathcal{A}}_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D703}}))$ .

We will need a partial answer to the question of when $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D706}},\operatorname{Loc}_{\unicode[STIX]{x1D706}}$ are mutually quasi-inverse equivalences (in this case we will say that abelian localization holds for $(\unicode[STIX]{x1D706},\unicode[STIX]{x1D703})$ ); see [Reference Braden, Proudfoot and WebsterBPW16, Corollary 5.17].

Let us proceed to translation equivalences. When $\unicode[STIX]{x1D712}\in \tilde{\mathfrak{p}}_{\mathbb{Z}}$ , we have an equivalence $\operatorname{Coh}({\mathcal{A}}_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D703}})\cong \operatorname{Coh}({\mathcal{A}}_{\unicode[STIX]{x1D706}+\unicode[STIX]{x1D712}}^{\unicode[STIX]{x1D703}})$ . This equivalence is given by tensoring with the ${\mathcal{A}}_{\unicode[STIX]{x1D706}+\unicode[STIX]{x1D712}}^{\unicode[STIX]{x1D703}}$ - ${\mathcal{A}}_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D703}}$ -bimodule to be denoted by ${\mathcal{A}}_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D712}}^{\unicode[STIX]{x1D703}}$ . This bimodule is the unique quantization of the line bundle ${\mathcal{O}}(\unicode[STIX]{x1D712})$ corresponding to $\unicode[STIX]{x1D712}$ . Similarly, we can define the $\tilde{{\mathcal{A}}}^{\unicode[STIX]{x1D703}}$ -bimodule $\tilde{{\mathcal{A}}}_{\tilde{\mathfrak{p}},\unicode[STIX]{x1D712}}^{\unicode[STIX]{x1D703}}$ quantizing the extension to $\tilde{X}$ of the line bundle corresponding to $\unicode[STIX]{x1D712}$ [Reference Braden, Proudfoot and WebsterBPW16, § 5.1]. Clearly, ${\mathcal{A}}_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D712}}^{\unicode[STIX]{x1D703}}$ is the specialization of $\tilde{{\mathcal{A}}}_{\tilde{\mathfrak{p}},\unicode[STIX]{x1D712}}^{\unicode[STIX]{x1D703}}$ .

We write ${\mathcal{A}}_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D712}}^{(\unicode[STIX]{x1D703})},\tilde{{\mathcal{A}}}_{\tilde{\mathfrak{p}},\unicode[STIX]{x1D712}}^{(\unicode[STIX]{x1D703})}$ for the global sections of ${\mathcal{A}}_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D712}}^{\unicode[STIX]{x1D703}},\tilde{{\mathcal{A}}}_{\tilde{\mathfrak{p}},\unicode[STIX]{x1D712}}^{\unicode[STIX]{x1D703}}$ , respectively. It is not known, in general, whether the bimodule ${\mathcal{A}}_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D712}}^{(\unicode[STIX]{x1D703})}$ is independent of $\unicode[STIX]{x1D703}$ . For an affine subspace $\mathfrak{p}\subset \tilde{\mathfrak{p}}$ , we set ${\mathcal{A}}_{\mathfrak{p},\unicode[STIX]{x1D712}}^{(\unicode[STIX]{x1D703})}:=\unicode[STIX]{x1D6E4}({\mathcal{A}}_{\tilde{\mathfrak{p}},\unicode[STIX]{x1D712}}^{\unicode[STIX]{x1D703}}\,\otimes _{\mathbb{C}[\tilde{\mathfrak{p}}]}\mathbb{C}[\mathfrak{p}])$ .

The proof repeats that of [Reference Bezrukavnikov and LosevBL13, Lemma 5.6(1)].

Let us provide a sufficient condition on $z\in \mathbb{C}$ for the algebras ${\mathcal{A}}_{\unicode[STIX]{x1D706}+z\unicode[STIX]{x1D712}}$ to be simple, to have finite homological dimension and also for the localization theorem to hold.

Proof. Let us prove (1); the proof is similar in spirit to those of [Reference Braden, Proudfoot and WebsterBPW16, § 5.3]. The localization holds for all parameters of the form $\unicode[STIX]{x1D706}+(z+n)\unicode[STIX]{x1D712}$ with $n\in \mathbb{Z}_{{\geqslant}0}$ if and only if the bimodules ${\mathcal{A}}_{\unicode[STIX]{x1D706}+(z+n)\unicode[STIX]{x1D712},\unicode[STIX]{x1D712}}^{(\unicode[STIX]{x1D703})},{\mathcal{A}}_{\ell ,-\unicode[STIX]{x1D712}}^{(\unicode[STIX]{x1D703})}|_{\unicode[STIX]{x1D706}+(z+n+1)\unicode[STIX]{x1D712}}$ define mutually inverse Morita equivalences; see [Reference Braden, Proudfoot and WebsterBPW16, § 5.3]. Note that by Lemma 2.3, ${\mathcal{A}}_{\unicode[STIX]{x1D706}+(z+n)\unicode[STIX]{x1D712},\unicode[STIX]{x1D712}}^{(\unicode[STIX]{x1D703})}={\mathcal{A}}_{\ell ,\unicode[STIX]{x1D712}}^{(\unicode[STIX]{x1D703})}|_{\unicode[STIX]{x1D706}+(z+n)\unicode[STIX]{x1D712}}$ .

The set, say $Z$ , of $z\in \mathbb{C}$ such that ${\mathcal{A}}_{\unicode[STIX]{x1D706}+z\unicode[STIX]{x1D712},\unicode[STIX]{x1D712}}^{(\unicode[STIX]{x1D703})}$ and ${\mathcal{A}}_{\ell ,-\unicode[STIX]{x1D712}}^{(\unicode[STIX]{x1D703})}|_{\unicode[STIX]{x1D706}+(z+1)\unicode[STIX]{x1D712}}$ are mutually inverse Morita equivalences is Zariski open. The reason is that $\{\unicode[STIX]{x1D706}+z\unicode[STIX]{x1D712}\mid z\in Z\}$ is the locus $\{\unicode[STIX]{x1D706}+z\unicode[STIX]{x1D712},z\in \mathbb{C}\}$ , where the natural homomorphisms (given by multiplication)

$$\begin{eqnarray}{\mathcal{A}}_{\ell +\unicode[STIX]{x1D712},-\unicode[STIX]{x1D712}}^{(\unicode[STIX]{x1D703})}\,\otimes _{{\mathcal{A}}_{\ell +\unicode[STIX]{x1D712}}}{\mathcal{A}}_{\ell ,\unicode[STIX]{x1D712}}^{(\unicode[STIX]{x1D703})}\rightarrow {\mathcal{A}}_{\ell },{\mathcal{A}}_{\ell ,\unicode[STIX]{x1D712}}^{(\unicode[STIX]{x1D703})}\,\otimes _{{\mathcal{A}}_{\ell }}{\mathcal{A}}_{\ell +\unicode[STIX]{x1D712},-\unicode[STIX]{x1D712}}^{(\unicode[STIX]{x1D703})}\rightarrow {\mathcal{A}}_{\ell +\unicode[STIX]{x1D712}}\end{eqnarray}$$

are isomorphisms. This locus is open by Proposition 2.6 below. It is non-empty by [Reference Braden, Proudfoot and WebsterBPW16, § 5.3]. Let $z_{1},\ldots ,z_{k}$ be the points in $Z$ . So, we see that localization holds for $\unicode[STIX]{x1D706}+z\unicode[STIX]{x1D712}$ if and only if $(z+\mathbb{Z}_{{\geqslant}0})\,\cap \,\mathbb{Z}=\varnothing$ . This finishes the proof of (1).

Let us prove (2): ${\mathcal{A}}_{\unicode[STIX]{x1D706}+z\unicode[STIX]{x1D712}}$ is simple for a Weil generic $z$ . Equivalently, we need to show that the ${\mathcal{A}}_{\unicode[STIX]{x1D706}+z\unicode[STIX]{x1D712}}\otimes {\mathcal{A}}_{-\unicode[STIX]{x1D706}-z\unicode[STIX]{x1D712}}$ -module ${\mathcal{A}}_{\unicode[STIX]{x1D706}+z\unicode[STIX]{x1D712}}$ is simple. By (1), abelian localization holds for the parameter $(\unicode[STIX]{x1D706}+z\unicode[STIX]{x1D712},-\unicode[STIX]{x1D706}+(-z)\unicode[STIX]{x1D712})$ and the variety $X^{\unicode[STIX]{x1D703}}\times X^{\unicode[STIX]{x1D703}}$ when $z$ is Weil generic. The module ${\mathcal{A}}_{\unicode[STIX]{x1D706}+z\unicode[STIX]{x1D712}}$ localizes to ${\mathcal{A}}_{\unicode[STIX]{x1D706}+z\unicode[STIX]{x1D712}}^{\unicode[STIX]{x1D703}}$ . The latter is simple because it is supported on the diagonal in $X^{\unicode[STIX]{x1D703}}\times X^{\unicode[STIX]{x1D703}}$ and has rank 1. This finishes the proof of simplicity of ${\mathcal{A}}_{\unicode[STIX]{x1D706}+z\unicode[STIX]{x1D712}}$ for a Weil generic $z$ .

Let us proceed to (3). The homological dimension of ${\mathcal{A}}_{\unicode[STIX]{x1D706}+z\unicode[STIX]{x1D712}}^{\unicode[STIX]{x1D703}}$ does not exceed that of $\operatorname{gr}{\mathcal{A}}_{\unicode[STIX]{x1D706}+z\unicode[STIX]{x1D712}}^{\unicode[STIX]{x1D703}}={\mathcal{O}}_{X}$ . The homological dimension of the latter does not exceed $\dim X$ (as any quasicoherent sheaf has injective resolution of length at most $\dim X$ ). The algebra ${\mathcal{A}}_{\unicode[STIX]{x1D706}+z\unicode[STIX]{x1D712}}$ has homological dimension equal to that of ${\mathcal{A}}_{\unicode[STIX]{x1D706}+z\unicode[STIX]{x1D712}}^{\pm \unicode[STIX]{x1D703}}$ provided localization holds for $(\unicode[STIX]{x1D706}+z\unicode[STIX]{x1D712},\pm \unicode[STIX]{x1D703})$ . Now our claim easily follows from (1) applied to both these situations.◻

We will also need the following general lemma, whose proof is completely analogous to that of [Reference Bezrukavnikov and LosevBL13, Lemma 5.7].

2.4 Harish-Chandra bimodules

Let us introduce the notion of a Harish-Chandra, shortly, a HC bimodule. Let ${\mathcal{A}},{\mathcal{A}}^{\prime }$ be two $\mathbb{Z}_{{\geqslant}0}$ -filtered algebras. Suppose we have a fixed isomorphism $\operatorname{gr}{\mathcal{A}}\cong \operatorname{gr}{\mathcal{A}}^{\prime }$ of graded algebras. Further, suppose that $A:=\operatorname{gr}{\mathcal{A}}=\operatorname{gr}{\mathcal{A}}^{\prime }$ is finitely generated and commutative. Let ${\mathcal{B}}$ be an ${\mathcal{A}}^{\prime }$ - ${\mathcal{A}}$ -bimodule. We say that ${\mathcal{B}}$ is HC if it is equipped with a bimodule filtration (called good) such that $\operatorname{gr}{\mathcal{B}}$ is a finitely generated $A$ -module (meaning that the left and the right actions of $A$ coincide). Note that a HC bimodule is automatically finitely generated as a left and as a right module.

An example is as follows: let ${\mathcal{A}}:={\mathcal{A}}_{\unicode[STIX]{x1D706}},{\mathcal{A}}^{\prime }:={\mathcal{A}}_{\unicode[STIX]{x1D706}+\unicode[STIX]{x1D712}}$ and ${\mathcal{B}}:={\mathcal{A}}_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D712}}^{(\unicode[STIX]{x1D703})}$ . Then ${\mathcal{B}}$ is HC; see [Reference Braden, Proudfoot and WebsterBPW16, § 6.3].

Below we will need a result about the supports of HC bimodules in the parameter spaces. Namely, suppose that $\mathfrak{p}$ is a finite-dimensional vector space and let ${\mathcal{A}}$ be a filtered $\mathbb{C}[\mathfrak{p}]$ -algebra such that $\mathfrak{p}$ has degree $d>0$ and $[{\mathcal{A}}_{{\leqslant}i},{\mathcal{A}}_{{\leqslant}j}]\subset {\mathcal{A}}_{{\leqslant}i+j-d}$ . Further, suppose that $\operatorname{gr}{\mathcal{A}}/(\mathfrak{p})$ is finitely generated and the corresponding Poisson scheme $X_{0}$ has finitely many symplectic leaves.

Now let ${\mathcal{B}}$ be a Harish-Chandra ${\mathcal{A}}$ -bimodule. Define its right $\mathfrak{p}$ -support $\operatorname{Supp}_{\mathfrak{p}}^{r}({\mathcal{B}})$ as the set of all $p\in \mathfrak{p}$ such that ${\mathcal{B}}\,\otimes _{\mathbb{C}[\mathfrak{p}]}\mathbb{C}_{p}\neq 0$ . Similarly, we can define the $\mathfrak{p}$ -support of $\operatorname{gr}{\mathcal{B}}$ (where the associated graded algebra is taken with respect to a good filtration) to be denoted by $\operatorname{Supp}_{\mathfrak{p}}(\operatorname{gr}{\mathcal{B}})$ . The following proposition generalizes [Reference Bezrukavnikov and LosevBL13, Proposition 5.15].

3.1 Definition

Let $\mathbb{K}$ be a field. Let ${\mathcal{C}}$ be a $\mathbb{K}$ -linear abelian category equivalent to the category of finite-dimensional modules over a split unital associative finite-dimensional $\mathbb{K}$ -algebra. We will write ${\mathcal{T}}$ for an indexing set for the simple objects of ${\mathcal{C}}$ . Let us write $L(\unicode[STIX]{x1D70F})$ for the simple object indexed by $\unicode[STIX]{x1D70F}\in {\mathcal{T}}$ and $P(\unicode[STIX]{x1D70F})$ for the projective cover of $L(\unicode[STIX]{x1D70F})$ .

The additional structure of a standardly stratified category on ${\mathcal{C}}$ is a partial pre-order ${\leqslant}$ on ${\mathcal{T}}$ that should satisfy axioms (SS1) and (SS2) to be explained below. Let us write $\unicode[STIX]{x1D6EF}$ for the set of equivalence classes of ${\leqslant}$ ; this is a poset (with partial order again denoted by ${\leqslant}$ ) that comes with a natural surjection $\unicode[STIX]{x1D71A}:{\mathcal{T}}{\twoheadrightarrow}\unicode[STIX]{x1D6EF}$ . The pre-order ${\leqslant}$ defines a filtration on ${\mathcal{C}}$ by Serre subcategories indexed by $\unicode[STIX]{x1D6EF}$ . Namely, to $\unicode[STIX]{x1D709}\in \unicode[STIX]{x1D6EF}$ we assign the subcategory ${\mathcal{C}}_{{\leqslant}\unicode[STIX]{x1D709}}$ that is the Serre span of the simples $L(\unicode[STIX]{x1D70F})$ with $\unicode[STIX]{x1D71A}(\unicode[STIX]{x1D70F})\leqslant \unicode[STIX]{x1D709}$ . Define ${\mathcal{C}}_{{<}\unicode[STIX]{x1D709}}$ analogously and let ${\mathcal{C}}_{\unicode[STIX]{x1D709}}$ denote the quotient ${\mathcal{C}}_{{\leqslant}\unicode[STIX]{x1D709}}/{\mathcal{C}}_{{<}\unicode[STIX]{x1D709}}$ . Let $\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D709}}$ denote the quotient functor ${\mathcal{C}}_{{\leqslant}\unicode[STIX]{x1D709}}{\twoheadrightarrow}{\mathcal{C}}_{\unicode[STIX]{x1D709}}$ . Let us write $\unicode[STIX]{x1D6E5}_{\unicode[STIX]{x1D709}}:{\mathcal{C}}_{\unicode[STIX]{x1D709}}\rightarrow {\mathcal{C}}_{{\leqslant}\unicode[STIX]{x1D709}}$ for the left adjoint functor of $\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D709}}$ ; it exists because ${\mathcal{C}}_{{\leqslant}\unicode[STIX]{x1D709}}$ is equivalent to the category of modules over a finite-dimensional associative algebra and $\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D709}}$ is a quotient functor (and so is given by taking Hom from a suitable bimodule). Also, we write $\operatorname{gr}{\mathcal{C}}$ for $\bigoplus _{\unicode[STIX]{x1D709}}{\mathcal{C}}_{\unicode[STIX]{x1D709}},\unicode[STIX]{x1D6E5}$ for $\bigoplus _{\unicode[STIX]{x1D709}}\unicode[STIX]{x1D6E5}_{\unicode[STIX]{x1D709}}:\operatorname{gr}{\mathcal{C}}\rightarrow {\mathcal{C}}$ . Finally, for $\unicode[STIX]{x1D70F}\in \unicode[STIX]{x1D71A}^{-1}(\unicode[STIX]{x1D709})$ , we write $L_{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D70F})$ for $\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D709}}(L(\unicode[STIX]{x1D70F}))$ , $P_{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D70F})$ for the projective cover of $L_{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D70F})$ in ${\mathcal{C}}_{\unicode[STIX]{x1D709}}$ and $\unicode[STIX]{x1D6E5}(\unicode[STIX]{x1D70F})$ for $\unicode[STIX]{x1D6E5}_{\unicode[STIX]{x1D709}}(P_{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D70F}))$ . The object $\unicode[STIX]{x1D6E5}(\unicode[STIX]{x1D70F})$ is called standard. The object $\overline{\unicode[STIX]{x1D6E5}}(\unicode[STIX]{x1D70F}):=\unicode[STIX]{x1D6E5}_{\unicode[STIX]{x1D709}}(L_{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D70F}))$ is called proper standard.

The axioms to be satisfied by $({\mathcal{C}},\leqslant )$ in order to give a standardly stratified structure are as follows.

1. (SS1) The functor $\unicode[STIX]{x1D6E5}:\operatorname{gr}{\mathcal{C}}\rightarrow {\mathcal{C}}$ is exact.

2. (SS2) The kernel of $P(\unicode[STIX]{x1D70F}){\twoheadrightarrow}\unicode[STIX]{x1D6E5}(\unicode[STIX]{x1D70F})$ has a filtration with successive quotients $\unicode[STIX]{x1D6E5}(\unicode[STIX]{x1D70F}^{\prime })$ , where $\unicode[STIX]{x1D70F}^{\prime }>\unicode[STIX]{x1D70F}$ .

Note that (SS1) allows us to identify $K_{0}(\operatorname{gr}{\mathcal{C}})$ and $K_{0}({\mathcal{C}})$ by means of $\unicode[STIX]{x1D6E5}$ . If (SS2) holds as well, then we also have the identification of $K_{0}(\operatorname{gr}{\mathcal{C}}\text{-}\!\operatorname{proj})$ and $K_{0}({\mathcal{C}}\text{-}\!\operatorname{proj})$ .

If all quotient categories ${\mathcal{C}}_{\unicode[STIX]{x1D709}}$ are equivalent to $\operatorname{Vect}$ , then we recover the classical definition of a highest-weight category. On the opposite end, if we take the trivial pre-order on ${\mathcal{T}}$ , then there is no additional structure.

3.2 (Proper) standardly filtered objects

We say that an object in ${\mathcal{C}}$ is standardly filtered if it admits a filtration whose successive quotients are standard. The notion of a proper standardly filtered object is introduced in a similar fashion. The categories of the standardly filtered objects and of the proper standardly filtered objects will be denoted by ${\mathcal{C}}^{\unicode[STIX]{x1D6E5}}$ and ${\mathcal{C}}^{\overline{\unicode[STIX]{x1D6E5}}}$ . Note that (SS1) implies that ${\mathcal{C}}^{\unicode[STIX]{x1D6E5}}\subset {\mathcal{C}}^{\overline{\unicode[STIX]{x1D6E5}}}$ .

Also, note that in a standardly stratified category the following hold:

(3.1) $$\begin{eqnarray}\displaystyle & \operatorname{Ext}_{{\mathcal{C}}}^{i}(\unicode[STIX]{x1D6E5}_{\unicode[STIX]{x1D709}}(M),\unicode[STIX]{x1D6E5}_{\unicode[STIX]{x1D709}^{\prime }}(N))\neq 0\Rightarrow \unicode[STIX]{x1D709}^{\prime }\leqslant \unicode[STIX]{x1D709}, & \displaystyle\end{eqnarray}$$

(3.2) $$\begin{eqnarray}\displaystyle & \operatorname{Ext}_{{\mathcal{C}}}^{i}(\unicode[STIX]{x1D6E5}_{\unicode[STIX]{x1D709}}(M),\unicode[STIX]{x1D6E5}_{\unicode[STIX]{x1D709}}(N))=\operatorname{Ext}_{{\mathcal{C}}_{\unicode[STIX]{x1D709}}}^{i}(M,N). & \displaystyle\end{eqnarray}$$

3.3 Subcategories and quotients

Suppose that ${\mathcal{C}}$ is standardly stratified.

Let $\unicode[STIX]{x1D6EF}_{0}$ be a poset ideal in $\unicode[STIX]{x1D6EF}$ . Let ${\mathcal{C}}_{\unicode[STIX]{x1D6EF}_{0}}$ denote the Serre span of the simples $L(\unicode[STIX]{x1D70F})$ with $\unicode[STIX]{x1D71A}(\unicode[STIX]{x1D70F})\in \unicode[STIX]{x1D6EF}_{0}$ . Then ${\mathcal{C}}_{\unicode[STIX]{x1D6EF}_{0}}$ is a standardly stratified category with pre-order on ${\mathcal{T}}_{0}:=\unicode[STIX]{x1D71A}^{-1}(\unicode[STIX]{x1D6EF}_{0})$ restricted from $\unicode[STIX]{x1D6EF}$ . Note that, for $\unicode[STIX]{x1D70F}\in {\mathcal{T}}_{0}$ , we have $\overline{\unicode[STIX]{x1D6E5}}_{\unicode[STIX]{x1D6EF}_{0}}(\unicode[STIX]{x1D70F})=\overline{\unicode[STIX]{x1D6E5}}(\unicode[STIX]{x1D70F}),\unicode[STIX]{x1D6E5}_{\unicode[STIX]{x1D6EF}_{0}}(\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D6E5}(\unicode[STIX]{x1D70F})$ , where the subscript $\unicode[STIX]{x1D6EF}_{0}$ refers to the objects computed in ${\mathcal{C}}_{\unicode[STIX]{x1D6EF}_{0}}$ .

The embedding $\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D6EF}_{0}}:{\mathcal{C}}_{\unicode[STIX]{x1D6EF}_{0}}^{\overline{\unicode[STIX]{x1D6E5}}}{\hookrightarrow}{\mathcal{C}}^{\overline{\unicode[STIX]{x1D6E5}}}$ admits a left adjoint functor $\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D6EF}_{0}}^{!}$ : to an object $M\in {\mathcal{C}}^{\overline{\unicode[STIX]{x1D6E5}}}$ , this functor assigns the maximal quotient lying in ${\mathcal{C}}_{\unicode[STIX]{x1D6EF}_{0}}^{\overline{\unicode[STIX]{x1D6E5}}}$ .

The following lemma will be of importance in the sequel.

Proof. Let $P_{\bullet }$ be a projective resolution (in ${\mathcal{C}}$ ) of $M\in {\mathcal{C}}_{\unicode[STIX]{x1D6EF}_{0}}$ . Then $\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D6EF}_{0}}^{!}P_{\bullet }$ is easily seen to be exact. So, it is a projective resolution of $M$ in ${\mathcal{C}}_{\unicode[STIX]{x1D6EF}_{0}}$ . It follows that for $M,N\in {\mathcal{C}}_{\unicode[STIX]{x1D6EF}_{0}}$ , we have

$$\begin{eqnarray}\operatorname{Ext}_{{\mathcal{C}}}^{i}(M,N)=\operatorname{Ext}_{{\mathcal{C}}_{\unicode[STIX]{x1D6EF}_{0}}}^{i}(M,N).\end{eqnarray}$$

This completes the proof.◻

Now let ${\mathcal{C}}^{\unicode[STIX]{x1D6EF}_{0}}$ be the quotient category ${\mathcal{C}}/{\mathcal{C}}_{\unicode[STIX]{x1D6EF}_{0}}$ . Let $\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D6EF}_{0}}$ denote the quotient functor ${\mathcal{C}}\rightarrow {\mathcal{C}}^{\unicode[STIX]{x1D6EF}_{0}}$ and let $\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D6EF}_{0}}^{!}$ be its left adjoint. The category ${\mathcal{C}}^{\unicode[STIX]{x1D6EF}_{0}}$ is standardly stratified with pre-order on ${\mathcal{T}}^{0}:={\mathcal{T}}\setminus {\mathcal{T}}_{0}$ restricted from ${\mathcal{T}}$ . For $\unicode[STIX]{x1D709}\in \unicode[STIX]{x1D6EF}^{0}:=\unicode[STIX]{x1D6EF}\setminus \unicode[STIX]{x1D6EF}_{0}$ , let $\unicode[STIX]{x1D6E5}_{\unicode[STIX]{x1D709}}^{0}$ denote the standardization functor for ${\mathcal{C}}^{\unicode[STIX]{x1D6EF}_{0}}$ . Then we have $\unicode[STIX]{x1D6E5}_{\unicode[STIX]{x1D709}}^{0}=\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D6EF}_{0}}\,\circ \,\unicode[STIX]{x1D6E5}_{\unicode[STIX]{x1D709}}$ . Let us also point out that $\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D6EF}_{0}}^{!}$ defines a full embedding $({\mathcal{C}}^{\unicode[STIX]{x1D6EF}_{0}})^{\overline{\unicode[STIX]{x1D6E5}}}{\hookrightarrow}{\mathcal{C}}^{\overline{\unicode[STIX]{x1D6E5}}}$ whose image coincides with the full subcategory ${\mathcal{C}}^{\overline{\unicode[STIX]{x1D6E5}},{\mathcal{T}}^{0}}$ consisting of all objects that admit a filtration with successive quotients $\unicode[STIX]{x1D6E5}(\unicode[STIX]{x1D70F})$ with $\unicode[STIX]{x1D70F}\in {\mathcal{T}}^{0}$ . This embedding sends $\unicode[STIX]{x1D6E5}^{0}(\unicode[STIX]{x1D70F})$ to $\unicode[STIX]{x1D6E5}(\unicode[STIX]{x1D70F})$ , $P^{0}(\unicode[STIX]{x1D70F})$ to $P(\unicode[STIX]{x1D70F})$ (here $\unicode[STIX]{x1D6E5}^{0}(\unicode[STIX]{x1D70F}),P^{0}(\unicode[STIX]{x1D70F})$ are the standard and projective objects in ${\mathcal{C}}^{\unicode[STIX]{x1D6EF}_{0}}$ labelled by $\unicode[STIX]{x1D70F}$ ).

3.4 Opposite category

Let ${\mathcal{C}}$ be a standardly stratified category. It turns out that the opposite category ${\mathcal{C}}^{\text{opp}}$ is also standardly stratified with the same pre-order ${\leqslant}$ ; see [Reference Losev and WebsterLW15, § 2.2]. The standard and proper standard objects for ${\mathcal{C}}^{\text{opp}}$ are denoted by $\unicode[STIX]{x1D6FB}(\unicode[STIX]{x1D70F})$ and $\overline{\unicode[STIX]{x1D6FB}}(\unicode[STIX]{x1D70F})$ . When viewed as objects of ${\mathcal{C}}$ , they are called costandard and proper costandard. The right adjoint functor to $\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D709}}$ will be denoted by $\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D709}}$ and we write $\unicode[STIX]{x1D6FB}$ for $\bigoplus _{\unicode[STIX]{x1D709}}\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D709}}$ . So, we have $\overline{\unicode[STIX]{x1D6FB}}(\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D6FB}(L_{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D70F}))$ and $\unicode[STIX]{x1D6FB}(\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D6FB}(I_{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D70F}))$ , where $I_{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D70F})$ is the injective envelope of $L_{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D70F})$ in ${\mathcal{C}}_{\unicode[STIX]{x1D709}}$ .

Let us write ${\mathcal{C}}^{\unicode[STIX]{x1D6FB}},{\mathcal{C}}^{\overline{\unicode[STIX]{x1D6FB}}}$ for the subcategories of costandardly filtered objects. We have the following standard lemma.

Let us also note the following fact.

3.5 Compatible standardly stratified structures

Now suppose that ${\mathcal{C}}$ is a highest-weight category with set of simples ${\mathcal{T}}$ and partial order ${\leqslant}$ . Consider a pre-order $\preccurlyeq$ on ${\mathcal{T}}$ refined by ${\leqslant}$ , meaning that $\unicode[STIX]{x1D70F}\leqslant \unicode[STIX]{x1D70F}^{\prime }$ implies that $\unicode[STIX]{x1D70F}\preccurlyeq \unicode[STIX]{x1D70F}^{\prime }$ . We say that $\preccurlyeq$ is standardly stratified if $({\mathcal{C}},\preccurlyeq )$ is a standardly stratified category. The corresponding standardly stratified structure will be called compatible with the initial highest-weight structure.

The following lemma gives a criterion for $\preccurlyeq$ to be standardly stratified. Namely, pick an equivalence class $\unicode[STIX]{x1D709}$ for $\preccurlyeq$ . Let $\unicode[STIX]{x1D71A}$ be the surjection from ${\mathcal{T}}$ to the set $\unicode[STIX]{x1D6EF}$ of equivalence classes for $\preccurlyeq$ . Set ${\mathcal{T}}_{0}:=\{\unicode[STIX]{x1D70F}\mid \unicode[STIX]{x1D71A}(\unicode[STIX]{x1D70F})\preccurlyeq \unicode[STIX]{x1D709}\}$ . Since ${\leqslant}$ refines $\preccurlyeq$ , we see that ${\mathcal{T}}_{0}$ is a poset ideal for ${\leqslant}$ . For the same reason, $\unicode[STIX]{x1D71A}^{-1}(\unicode[STIX]{x1D709})$ is an interval for ${\leqslant}$ . It follows that the subquotient ${\mathcal{C}}_{\unicode[STIX]{x1D709}}$ of ${\mathcal{C}}$ corresponding to $\unicode[STIX]{x1D709}$ has a natural highest-weight structure.

Let $\unicode[STIX]{x1D704}_{{\mathcal{T}}_{0}}:{\mathcal{C}}_{{\mathcal{T}}_{0}}{\hookrightarrow}{\mathcal{C}}$ be an inclusion. For a standardly filtered object $M\in {\mathcal{C}}$ , define an object $M(\unicode[STIX]{x1D709})\in {\mathcal{C}}_{\unicode[STIX]{x1D709}}$ as $\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D709}}\circ \unicode[STIX]{x1D704}_{{\mathcal{T}}_{0}}^{!}(M)$ . The object $M(\unicode[STIX]{x1D709})$ is standardly filtered.

Similarly, for a costandardly filtered object $N$ , we can define $N(\unicode[STIX]{x1D709})$ in a similar way (but we need to use the right adjoint $\unicode[STIX]{x1D704}_{{\mathcal{T}}_{0}}^{\ast }$ instead of the left adjoint $\unicode[STIX]{x1D704}_{{\mathcal{T}}_{0}}^{!}$ ).

Conversely, let ${\mathcal{C}}$ be a standardly stratified category with respect to a pre-order $\preccurlyeq$ . Suppose that each ${\mathcal{C}}_{\unicode[STIX]{x1D709}}$ is highest weight with respect to an order ${\leqslant}_{\unicode[STIX]{x1D709}}$ for all $\unicode[STIX]{x1D709}\in \unicode[STIX]{x1D6EF}$ . Then ${\mathcal{C}}$ is highest weight with respect to the order ${\leqslant}$ defined as follows: $\unicode[STIX]{x1D70F}\leqslant \unicode[STIX]{x1D70F}^{\prime }$ if and only if either $\unicode[STIX]{x1D70F}\prec \unicode[STIX]{x1D70F}^{\prime }$ or $\unicode[STIX]{x1D70F}\sim \unicode[STIX]{x1D70F}^{\prime }$ and $\unicode[STIX]{x1D70F}\leqslant _{\unicode[STIX]{x1D709}}\unicode[STIX]{x1D70F}^{\prime }$ for $\unicode[STIX]{x1D709}=\unicode[STIX]{x1D71A}(\unicode[STIX]{x1D70F})$ .

4.1 Cartan subquotient: algebra level

Let ${\mathcal{A}}=\bigcup _{i\in \mathbb{Z}}{\mathcal{A}}_{{\leqslant}i}$ be a filtered associative algebra (with a complete and separated filtration) such that $[{\mathcal{A}}_{{\leqslant}i},{\mathcal{A}}_{{\leqslant}j}]\subset {\mathcal{A}}_{{\leqslant}i+j-d}$ for some $d\in \mathbb{Z}_{{>}0}$ . The algebra $\operatorname{gr}{\mathcal{A}}$ is commutative and we require that it is finitely generated.

Suppose that ${\mathcal{A}}$ is equipped with a pro-rational Hamiltonian $\mathbb{C}^{\times }$ -action $\unicode[STIX]{x1D708}$ that preserves the filtration (pro-rational means that the action of $\mathbb{C}^{\times }$ on ${\mathcal{A}}/{\mathcal{A}}_{{\leqslant}i}$ is rational for all $i$ ). Let $h\in {\mathcal{A}}$ be the image of $1$ under the comoment map and let ${\mathcal{A}}^{i}$ denote the $i$ th graded component, so that ${\mathcal{A}}^{i}:=\{a\mid [h,a]=ia\}$ . We set ${\mathcal{A}}^{{\geqslant}0}:=\bigoplus _{i\geqslant 0}{\mathcal{A}}^{i},{\mathcal{A}}^{{>}0}:=\bigoplus _{i>0}{\mathcal{A}}^{i},\mathsf{C}_{\unicode[STIX]{x1D708}}({\mathcal{A}}):={\mathcal{A}}^{{\geqslant}0}/({\mathcal{A}}^{{\geqslant}0}\,\cap \,{\mathcal{A}}{\mathcal{A}}^{{>}0})$ . Note that ${\mathcal{A}}^{{\geqslant}0}\,\cap \,{\mathcal{A}}{\mathcal{A}}^{{>}0}$ is a two-sided ideal in ${\mathcal{A}}^{{\geqslant}0}$ . We have a natural isomorphism

(4.1) $$\begin{eqnarray}\mathsf{C}_{\unicode[STIX]{x1D708}}({\mathcal{A}})\cong {\mathcal{A}}^{0}\bigg/\bigoplus _{i>0}{\mathcal{A}}^{-i}{\mathcal{A}}^{i}.\end{eqnarray}$$

The algebra $\mathsf{C}_{\unicode[STIX]{x1D708}}({\mathcal{A}})$ inherits a filtration from ${\mathcal{A}}$ . This filtration is complete and separated as any two-sided ideal in ${\mathcal{A}}^{0}$ is closed with respect to the topology induced by the filtration; compare [Reference LosevLos11, Lemma 2.4.4]. The algebra $\operatorname{gr}\mathsf{C}_{\unicode[STIX]{x1D708}}({\mathcal{A}})$ is commutative and, being a quotient of $(\operatorname{gr}{\mathcal{A}})^{\unicode[STIX]{x1D708}(\mathbb{C}^{\times })}$ , finitely generated.

Let us note that the definitions of ${\mathcal{A}}^{{>}0},{\mathcal{A}}^{{\geqslant}0},\mathsf{C}_{\unicode[STIX]{x1D708}}({\mathcal{A}})$ make sense even if the action $\unicode[STIX]{x1D708}$ is not Hamiltonian.

4.2 Categories ${\mathcal{O}}$ : algebra level

Let ${\mathcal{A}}$ be as in § 4.1 and assume that the filtration mentioned there is by $\mathbb{Z}_{{\geqslant}0}$ . By the category ${\mathcal{O}}_{\unicode[STIX]{x1D708}}({\mathcal{A}})$ , we mean the full subcategory of the category ${\mathcal{A}}\text{-}\!\operatorname{mod}$ of the finitely generated ${\mathcal{A}}$ -modules consisting of all modules such that ${\mathcal{A}}^{{>}0}$ acts locally nilpotently. We have the Verma module functor $\unicode[STIX]{x1D6E5}_{\unicode[STIX]{x1D708}}:\mathsf{C}_{\unicode[STIX]{x1D708}}({\mathcal{A}})\text{-}\!\operatorname{mod}\rightarrow {\mathcal{O}}_{\unicode[STIX]{x1D708}}({\mathcal{A}})$ given by $\unicode[STIX]{x1D6E5}_{\unicode[STIX]{x1D708}}(N):={\mathcal{A}}\,\otimes _{{\mathcal{A}}^{{\geqslant}0}}N=({\mathcal{A}}/{\mathcal{A}}{\mathcal{A}}^{{>}0})\,\otimes _{\mathsf{C}_{\unicode[STIX]{x1D708}}({\mathcal{A}})}N$ . Note that if $h$ acts on $N$ with eigenvalue $\unicode[STIX]{x1D6FC}$ (so that $N$ is a single generalized $h$ -eigenspace), then $h$ acts locally finitely on $\unicode[STIX]{x1D6E5}_{\unicode[STIX]{x1D708}}(N)$ with eigenvalues in $\unicode[STIX]{x1D6FC}+\mathbb{Z}_{{\leqslant}0}$ . The generalized eigenspace with eigenvalue $\unicode[STIX]{x1D6FC}$ coincides with $({\mathcal{A}}/{\mathcal{A}}{\mathcal{A}}^{{>}0})^{\unicode[STIX]{x1D708}(\mathbb{C}^{\times })}\otimes _{\mathsf{C}_{\unicode[STIX]{x1D708}}({\mathcal{A}})}N=N$ (note that $\mathsf{C}_{\unicode[STIX]{x1D708}}({\mathcal{A}})$ is identified with  $({\mathcal{A}}/{\mathcal{A}}{\mathcal{A}}^{{>}0})^{\unicode[STIX]{x1D708}(\mathbb{C}^{\times })}$ ). The module $\unicode[STIX]{x1D6E5}_{\unicode[STIX]{x1D708}}(N)$ has the maximal submodule that does not intersect $N$ ; the quotient is denoted by $L_{\unicode[STIX]{x1D708}}(N)$ . Note that $L_{\unicode[STIX]{x1D708}}(N)$ is simple provided $N$ is simple. Conversely, any simple object $L$ in ${\mathcal{O}}_{\unicode[STIX]{x1D708}}({\mathcal{A}})$ is of the form $L_{\unicode[STIX]{x1D708}}(N)$ for a unique simple $N\in \mathsf{C}_{\unicode[STIX]{x1D708}}({\mathcal{A}})\text{-}\!\operatorname{mod}$ ; this simple $N$ coincides with the annihilator $L^{{\mathcal{A}}^{{>}0}}$ of ${\mathcal{A}}^{{>}0}$ in $L$ .

Assume now that $\dim \mathsf{C}_{\unicode[STIX]{x1D708}}({\mathcal{A}})<\infty$ . Let $N_{1},\ldots ,N_{r}$ be the full list of the simple $\mathsf{C}_{\unicode[STIX]{x1D708}}({\mathcal{A}})$ -modules. Let $\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{r}$ be the eigenvalues of $h$ on these modules (note that $h$ maps into the center of $\mathsf{C}_{\unicode[STIX]{x1D708}}({\mathcal{A}})$ ). We define a partial order ${\leqslant}$ on the set $N_{1},\ldots ,N_{r}$ by setting $N_{i}\leqslant N_{j}$ if $\unicode[STIX]{x1D6FC}_{j}-\unicode[STIX]{x1D6FC}_{i}\in \mathbb{Z}_{{>}0}$ or $i=j$ . Using the order, it is easy to prove the following.

Also, an argument similar to that in [Reference Ginzburg, Guay, Opdam and RouquierGGOR03, § 2.4] implies that ${\mathcal{O}}_{\unicode[STIX]{x1D708}}({\mathcal{A}})$ has enough projectives.

Now let us describe the opposite category. The opposite algebra ${\mathcal{A}}^{\text{opp}}$ is also graded. Consider the category ${\mathcal{O}}_{-\unicode[STIX]{x1D708}}({\mathcal{A}}^{\text{opp}})$ . By (4.1), we have a natural identification $\mathsf{C}_{-\unicode[STIX]{x1D708}}({\mathcal{A}}^{\text{opp}})\cong \mathsf{C}_{\unicode[STIX]{x1D708}}({\mathcal{A}})^{\text{opp}}$ (indeed, the $i$ th graded component for $\unicode[STIX]{x1D708}$ is the same as the $-i$ th graded component for $-\unicode[STIX]{x1D708}$ ). For a module $M\in {\mathcal{O}}_{\unicode[STIX]{x1D708}}({\mathcal{A}})$ , we define its restricted dual $M^{(\ast )}$ as $\bigoplus _{\unicode[STIX]{x1D6FC}}M_{\unicode[STIX]{x1D6FC}}^{\ast }$ , where $M_{\unicode[STIX]{x1D6FC}}$ stands for the generalized eigenspace for $h$ with eigenvalue $\unicode[STIX]{x1D6FC}$ . Note that $M^{(\ast )}\in {\mathcal{O}}_{-\unicode[STIX]{x1D708}}({\mathcal{A}}^{\text{opp}})$ by (4) of Lemma 4.1. Similarly, for $M^{\prime }\in {\mathcal{O}}_{-\unicode[STIX]{x1D708}}({\mathcal{A}}^{\text{opp}})$ , we can form ${M^{\prime }}^{(\ast )}$ and this will be an object of ${\mathcal{O}}_{\unicode[STIX]{x1D708}}({\mathcal{A}})$ . The functors of taking the restricted dual define mutually inverse contravariant equivalences between ${\mathcal{O}}_{\unicode[STIX]{x1D708}}({\mathcal{A}})$ and ${\mathcal{O}}_{-\unicode[STIX]{x1D708}}({\mathcal{A}}^{\text{opp}})$ .

Using these equivalences, one can introduce dual Verma modules $\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D708}}(N):=\unicode[STIX]{x1D6E5}_{-\unicode[STIX]{x1D708}}^{\text{opp}}(N^{\ast })^{(\ast )}\in {\mathcal{O}}_{\unicode[STIX]{x1D708}}({\mathcal{A}})$ . Equivalently, $\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D708}}(N)=\operatorname{Hom}_{\mathsf{C}_{\unicode[STIX]{x1D708}}({\mathcal{A}})}^{\text{restr}}({\mathcal{A}}/{\mathcal{A}}^{{<}0}{\mathcal{A}},N)$ . Here we take the restricted Hom, so that $\operatorname{Hom}_{\mathsf{C}_{\unicode[STIX]{x1D708}}({\mathcal{A}})}^{\text{restr}}({\mathcal{A}}/{\mathcal{A}}^{{<}0}{\mathcal{A}},N)=\bigoplus _{i}\operatorname{Hom}_{\mathsf{C}_{\unicode[STIX]{x1D708}}({\mathcal{A}})}(({\mathcal{A}}/{\mathcal{A}}^{{<}0}{\mathcal{A}})_{i},N)$ . Indeed, the functors $\operatorname{Hom}_{{\mathcal{O}}_{-\unicode[STIX]{x1D708}}({\mathcal{A}})}(\bullet ,\unicode[STIX]{x1D6E5}_{-\unicode[STIX]{x1D708}}^{\text{opp}}(N^{\ast })^{(\ast )})$ and $\operatorname{Hom}_{{\mathcal{O}}_{-\unicode[STIX]{x1D708}}({\mathcal{A}})}(\bullet ,\operatorname{Hom}_{\mathsf{C}_{\unicode[STIX]{x1D708}}({\mathcal{A}})}^{\text{restr}}({\mathcal{A}}/{\mathcal{A}}^{{<}0}{\mathcal{A}},N))$ are isomorphic.

4.3 Case of quantizations of symplectic resolutions

Now let $X$ be a conical symplectic resolution of an affine variety $X_{0}$ . Let ${\mathcal{A}}_{\unicode[STIX]{x1D706}}$ stand for the algebra of global sections of the filtered quantization ${\mathcal{A}}_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D703}}$ of $X$ corresponding to $\unicode[STIX]{x1D706}\in \tilde{\mathfrak{p}}:=H^{2}(X,\mathbb{C})$ . Assume that we have a Hamiltonian action $\unicode[STIX]{x1D708}$ of $\mathbb{C}^{\times }$ on $X$ with finitely many fixed points that commutes with the contracting $\mathbb{C}^{\times }$ -action. This action lifts to a Hamiltonian action on a quantization and hence gives rise to a Hamiltonian action on ${\mathcal{A}}_{\unicode[STIX]{x1D706}}$ again denoted by $\unicode[STIX]{x1D708}$ .

The following results were obtained in [Reference Braden, Licata, Proudfoot and WebsterBLPW16, § 5].

The image of the embedding in (3) is the category $D_{{\mathcal{O}}_{\unicode[STIX]{x1D708}}}^{b}({\mathcal{A}}_{\unicode[STIX]{x1D706}}\text{-}\!\operatorname{mod})$ of all complexes whose homologies lie in ${\mathcal{O}}_{\unicode[STIX]{x1D708}}({\mathcal{A}}_{\unicode[STIX]{x1D706}})$ .

We also have the category ${\mathcal{O}}$ on the level of sheaves. Following [Reference Braden, Licata, Proudfoot and WebsterBLPW16, § 3.3], consider the category ${\mathcal{O}}_{\unicode[STIX]{x1D708}}({\mathcal{A}}_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D703}})$ . It consists of all coherent ${\mathcal{A}}_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D703}}$ -modules that come with a good filtration stable under $h$ and that are supported on the contracting locus $Y$ for $\unicode[STIX]{x1D708}$ in $X$ , i.e., $Y:=\{x\in X\mid \lim _{t\rightarrow 0}\unicode[STIX]{x1D708}(t).x\text{ exists}\}$ . Note that $Y$ is a Lagrangian subvariety.

The translation equivalences preserve categories ${\mathcal{O}}_{\unicode[STIX]{x1D708}}$ and so do $\unicode[STIX]{x1D6E4}$ and $\operatorname{Loc}$ ; see [Reference Braden, Licata, Proudfoot and WebsterBLPW16, Lemma 3.17 and Corollary 3.19].

Again, we assume that $\unicode[STIX]{x1D708}$ is generic. Pick an ample class $\unicode[STIX]{x1D712}\in \operatorname{Pic}(X)$ . Using equivalences ${\mathcal{O}}_{\unicode[STIX]{x1D708}}({\mathcal{A}}_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D703}})\xrightarrow[{}]{{\sim}}{\mathcal{O}}_{\unicode[STIX]{x1D708}}({\mathcal{A}}_{\unicode[STIX]{x1D706}+n\unicode[STIX]{x1D712}}^{\unicode[STIX]{x1D703}})\xrightarrow[{}]{{\sim}}{\mathcal{O}}_{\unicode[STIX]{x1D708}}({\mathcal{A}}_{\unicode[STIX]{x1D706}+n\unicode[STIX]{x1D712}})$ for $n\gg 0$ , we can carry the highest-weight structure from ${\mathcal{O}}_{\unicode[STIX]{x1D708}}({\mathcal{A}}_{\unicode[STIX]{x1D706}+n\unicode[STIX]{x1D712}})$ to ${\mathcal{O}}_{\unicode[STIX]{x1D708}}({\mathcal{A}}_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D703}})$ . However, we can use a different partial order getting the same collection of standard objects. Namely, for $p\in X^{\unicode[STIX]{x1D708}(\mathbb{C}^{\times })}$ , let $Y_{p}$ denote the contracting component of $p$ , $Y_{p}:=\{x\in X\mid \lim _{t\rightarrow 0}\unicode[STIX]{x1D708}(t)x=p\}$ . Define the pre-order $p^{\prime }\preccurlyeq p$ on $X^{T}$ given by $p^{\prime }\preccurlyeq p$ if and only if $p^{\prime }\in \overline{Y_{p}}$ (we will see that this is indeed a pre-order in Lemma 6.1 that covers a more general situation below).

Define the partial order $\preccurlyeq _{\unicode[STIX]{x1D708}}$ on $X^{\unicode[STIX]{x1D708}(\mathbb{C}^{\times })}$ by taking the transitive closure of the relation $\preccurlyeq _{\unicode[STIX]{x1D708}}$ . Set $Y_{\preccurlyeq p}:=\bigcup _{p^{\prime }\preccurlyeq p}Y_{p^{\prime }}$ ; this is a closed Lagrangian subvariety in $X$ . Further, set

$$\begin{eqnarray}{\mathcal{O}}_{\unicode[STIX]{x1D708}}({\mathcal{A}}_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D703}})_{\preccurlyeq p}:=\{M\in {\mathcal{O}}_{\unicode[STIX]{x1D708}}({\mathcal{A}}_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D703}})\mid \operatorname{Supp}(M)\subset Y_{\preccurlyeq p}\}.\end{eqnarray}$$

Here, as usual, we write $\operatorname{Supp}(M)$ for the support of $\operatorname{gr}M$ in $X$ , where the associated graded algebra is taken with respect to a good filtration.

It turns out that ${\mathcal{O}}_{\unicode[STIX]{x1D708}}({\mathcal{A}}_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D703}})$ is a highest-weight category with respect to this order. The standard objects are localizations of Verma modules $\operatorname{Loc}_{\unicode[STIX]{x1D706}+n\unicode[STIX]{x1D712}}(\unicode[STIX]{x1D6E5}_{\unicode[STIX]{x1D708}}(N))\in {\mathcal{O}}_{\unicode[STIX]{x1D708}}({\mathcal{A}}_{\unicode[STIX]{x1D706}+n\unicode[STIX]{x1D712}}^{\unicode[STIX]{x1D703}})\cong {\mathcal{O}}_{\unicode[STIX]{x1D708}}({\mathcal{A}}_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D703}})$ ; see [Reference Braden, Licata, Proudfoot and WebsterBLPW16, § 5.3].

Below we will need to understand the structure of the category ${\mathcal{O}}_{\unicode[STIX]{x1D708}}({\mathcal{A}}_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D703}})$ in the case when $X$ splits into the product $X^{1}\times X^{2}$ of two conical symplectic resolutions. Both $\unicode[STIX]{x1D706}$ and $\unicode[STIX]{x1D703}$ are naturally pairs $(\unicode[STIX]{x1D706}^{1},\unicode[STIX]{x1D706}^{2}),(\unicode[STIX]{x1D703}^{1},\unicode[STIX]{x1D703}^{2})$ .

Proof. Note that we have

(4.2) $$\begin{eqnarray}\dim \text{Ext}_{{\mathcal{A}}_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D703}}}^{i}(P^{1}(p_{1})\boxtimes P^{2}(p_{2}),L^{1}(p_{1}^{\prime })\boxtimes L^{2}(p_{2}^{\prime }))=\unicode[STIX]{x1D6FF}_{i,0}\unicode[STIX]{x1D6FF}_{p_{1},p_{1}^{\prime }}\unicode[STIX]{x1D6FF}_{p_{2},p_{2}^{\prime }},\end{eqnarray}$$

where $p_{i},p_{i}^{\prime }\in X^{i\unicode[STIX]{x1D708}(\mathbb{C}^{\times })},i=1,2,$ and $P^{i},L^{i}$ stand for the projective and simple objects in the categories ${\mathcal{O}}_{\unicode[STIX]{x1D708}}({\mathcal{A}}_{\unicode[STIX]{x1D706}^{i}}^{i\unicode[STIX]{x1D703}^{i}})$ . Indeed, (4.2) is clear for the categories on the level of algebras and on the level of sheaves it follows from the localization argument. The objects $L^{1}(p_{1}^{\prime })\boxtimes L^{2}(p_{2}^{\prime })$ are the simples in ${\mathcal{O}}_{\unicode[STIX]{x1D708}}({\mathcal{A}}^{\unicode[STIX]{x1D703}})$ . It follows that the objects $P^{1}(p_{1})\boxtimes P^{2}(p_{2})$ are the indecomposable projectives.◻

4.4 Holonomic modules

In this section, we recall some results from [Reference LosevLos17]. Let $X_{0}$ be a Poisson variety with finitely many symplectic leaves. Following [Reference LosevLos17], we say that a subvariety $Y_{0}\subset X_{0}$ is isotropic if its intersection with every leaf is isotropic in the leaf. For example, when $X_{0}$ admits a symplectic resolution of singularities $(X,\unicode[STIX]{x1D70C})$ , then $Y_{0}\subset X_{0}$ is isotropic if and only if $\unicode[STIX]{x1D70C}^{-1}(Y_{0})$ is isotropic; see [Reference LosevLos17, Lemma 5.1].

Now let $X,X_{0},\unicode[STIX]{x1D708}$ be as in § 4.3. Let $Y_{0}\subset X_{0}$ be the contracting locus of $\unicode[STIX]{x1D708}$ . Then $\unicode[STIX]{x1D70C}^{-1}(Y_{0})=Y$ . We conclude that $Y_{0}\subset X_{0}$ is isotropic.

Let us proceed to holonomic ${\mathcal{A}}_{\unicode[STIX]{x1D706}}$ -modules. We say that a finitely generated ${\mathcal{A}}_{\unicode[STIX]{x1D706}}$ -module is holonomic if its support in $X_{0}$ is holonomic. In particular, any module in ${\mathcal{O}}_{\unicode[STIX]{x1D708}}({\mathcal{A}}_{\unicode[STIX]{x1D706}})$ is supported on $Y_{0}$ and so is holonomic.

For holonomic modules, we have the following result; see [Reference LosevLos17, Theorems 1.2 and 1.3].

We will need a consequence of this proposition and Lemma 2.4.

Let us provide one more example of holonomic modules. Let ${\mathcal{B}}$ be a HC ${\mathcal{A}}_{\unicode[STIX]{x1D706}}$ -bimodule. Then it is a holonomic ${\mathcal{A}}_{\unicode[STIX]{x1D706}}\otimes {\mathcal{A}}_{\unicode[STIX]{x1D706}}^{\text{opp}}$ -module (its support is the diagonal in $X_{0}\times X_{0}$ that is easily seen to be isotropic).

5.1 Hamiltonian actions and fixed point components

Suppose that we have a Hamiltonian action $\unicode[STIX]{x1D708}$ of $\mathbb{C}^{\times }$ on $X=X^{\unicode[STIX]{x1D703}}$ that commutes with the contracting action of $\mathbb{C}^{\times }$ . By the universality of the deformations, the Hamiltonian action extends to both $\tilde{X}^{\unicode[STIX]{x1D703}}$ and $\tilde{{\mathcal{A}}}^{\unicode[STIX]{x1D703}}$ . Since $H_{DR}^{1}(X)=0$ , we see that the actions on $\tilde{X}^{\unicode[STIX]{x1D703}},\tilde{{\mathcal{A}}}^{\unicode[STIX]{x1D703}}$ are Hamiltonian as well. For the action on $\tilde{{\mathcal{A}}}^{\unicode[STIX]{x1D703}}$ , this means, by definition, that there is a $\unicode[STIX]{x1D708}(\mathbb{C}^{\times })$ -invariant element $h\in \tilde{{\mathcal{A}}}$ such that the derivation $[h,\cdot ]$ of $\tilde{{\mathcal{A}}}^{\unicode[STIX]{x1D703}}$ coincides with the derivation induced by the $\mathbb{C}^{\times }$ -action.

We also note that the irreducible components of $X_{\unicode[STIX]{x1D712}}^{\unicode[STIX]{x1D708}(\mathbb{C}^{\times })}$ are in a natural bijection with those of $X^{\unicode[STIX]{x1D708}(\mathbb{C}^{\times })}$ .

5.2 Cartan subquotient: sheaf level

We start with a symplectic variety $X$ equipped with a $\mathbb{C}^{\times }$ -action that rescales the symplectic form and also with a commuting Hamiltonian action $\unicode[STIX]{x1D708}$ . Of course, it still makes sense to speak about quantizations of $X$ that are Hamiltonian for $\unicode[STIX]{x1D708}$ . We want to construct a quantization $\mathsf{C}_{\unicode[STIX]{x1D708}}({\mathcal{A}}^{\unicode[STIX]{x1D703}})$ of $X^{\unicode[STIX]{x1D708}(\mathbb{C}^{\times })}$ starting from a Hamiltonian quantization ${\mathcal{A}}^{\unicode[STIX]{x1D703}}$ of $X$ .

The variety $X$ can be covered by $(\mathbb{C}^{\times })^{2}$ -stable open affine subvarieties. Pick such a subvariety $X^{\prime }$ with $(X^{\prime })^{\unicode[STIX]{x1D708}(\mathbb{C}^{\times })}\neq \varnothing$ . We remark that the open subsets of the form $(X^{\prime })^{\unicode[STIX]{x1D708}(\mathbb{C}^{\times })}$ form a base of the Zariski topology on $X^{\unicode[STIX]{x1D708}(\mathbb{C}^{\times })}$ .

The following proposition defines the sheaf $\mathsf{C}_{\unicode[STIX]{x1D708}}({\mathcal{A}}^{\unicode[STIX]{x1D703}})$ .

5.3 Comparison between algebra and sheaf levels

Now let us assume that $X$ is a conical symplectic resolution of $X_{0}$ . We write ${\mathcal{A}}_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D703}}$ for the quantization of $X$ corresponding to $\unicode[STIX]{x1D706}$ and ${\mathcal{A}}_{\unicode[STIX]{x1D706}}$ for its algebra of global sections. By the construction, for any $\unicode[STIX]{x1D706}\in H^{2}(X)$ , there are a natural homomorphism $\mathsf{C}_{\unicode[STIX]{x1D708}}({\mathcal{A}}_{\unicode[STIX]{x1D706}})\rightarrow \unicode[STIX]{x1D6E4}(\mathsf{C}_{\unicode[STIX]{x1D708}}({\mathcal{A}}_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D703}}))$ as well as a natural homomorphism $\mathsf{C}_{\unicode[STIX]{x1D708}}(\mathbb{C}[X_{0}])\rightarrow \unicode[STIX]{x1D6E4}(\mathsf{C}_{\unicode[STIX]{x1D708}}({\mathcal{O}}_{X}))=\mathbb{C}[X^{\unicode[STIX]{x1D708}(\mathbb{C}^{\times })}]$ . Our goal in this section is to prove the following result.

In the proof, we will need a lemma describing some properties of the natural morphism $X^{\unicode[STIX]{x1D708}(\mathbb{C}^{\times })}\rightarrow \operatorname{Spec}(\mathsf{C}_{\unicode[STIX]{x1D708}}(\mathbb{C}[X_{0}]))$ .

5.4 Correspondence between parameters

Our next goal is to understand how to recover the periods of the direct summands $\mathsf{C}_{\unicode[STIX]{x1D708}}({\mathcal{A}}^{\unicode[STIX]{x1D703}})$ from that of ${\mathcal{A}}^{\unicode[STIX]{x1D703}}$ . We will assume that $H^{i}(X^{\unicode[STIX]{x1D708}(\mathbb{C}^{\times })},{\mathcal{O}})=0$ for $i>0$ , but we will not require that of $X$ . The period map still makes sense for $X$ ; see [Reference Bezrukavnikov and KaledinBK04, § 4]. Consider the decomposition $X^{\unicode[STIX]{x1D708}(\mathbb{C}^{\times })}=\bigsqcup _{i}X_{i}^{0}$ into connected components. Let $Y_{i}$ denote the contracting locus of $X_{i}^{0}$ and let ${\mathcal{A}}_{i}^{\unicode[STIX]{x1D703}0}$ be the restriction of $\mathsf{C}_{\unicode[STIX]{x1D708}}({\mathcal{A}}^{\unicode[STIX]{x1D703}})$ to $X_{i}^{0}$ . To determine the period of ${\mathcal{A}}_{i}^{\unicode[STIX]{x1D703}0}$ , we will quantize $Y_{i}$ and then use results from [Reference Baranovsky, Ginzburg, Kaledin and PecharichBGKP16] on quantizations of line bundles on Lagrangian subvarieties.

First of all, let us consider the case when $X$ is affine and so is quantized by a single algebra,  ${\mathcal{A}}$ . We will quantize the contracting locus $Y$ by a single ${\mathcal{A}}$ - $\mathsf{C}_{\unicode[STIX]{x1D708}}({\mathcal{A}})$ -bimodule; this bimodule is ${\mathcal{A}}/{\mathcal{A}}{\mathcal{A}}^{{>}0}$ .

Now let us consider the non-affine case. Let us cover $X\setminus \bigcup _{k\neq i}X_{k}^{0}$ with $(\mathbb{C}^{\times })^{2}$ -stable open affine subsets $X^{j}$ . We may assume that $X^{j}\,\cap \,Y_{i}=\varnothing$ or $X^{j}\,\cap \,Y_{i}=\unicode[STIX]{x1D70B}_{i}^{-1}(X^{j}\,\cap \,X_{i}^{0})$ , where $\unicode[STIX]{x1D70B}_{i}:Y_{i}\rightarrow X_{i}^{0}$ is the projection. For this, we first choose some covering by $(\mathbb{C}^{\times })^{2}$ -stable open affine subsets. Then we delete $Y_{i}\setminus \unicode[STIX]{x1D70B}_{i}^{-1}(X^{j}\,\cap \,X_{i}^{0})$ from each $X^{j}$ ; we still have a covering. We cover the remainder of each $X^{j}$ by subsets that are pre-images of open affine subsets on $X^{j}/\!/\unicode[STIX]{x1D708}(\mathbb{C}^{\times })$ that do not intersect $X_{j}^{0}/\!/\unicode[STIX]{x1D708}(\mathbb{C}^{\times })$ . It is easy to see that this covering has the required properties. Let us replace $X$ with the union of $X^{j}$ that intersect $Y_{i}$ .

After this replacement, we can quantize $Y_{i}$ by an ${\mathcal{A}}^{\unicode[STIX]{x1D703}}$ - $\mathsf{C}_{\unicode[STIX]{x1D708}}({\mathcal{A}}^{\unicode[STIX]{x1D703}})$ -bimodule. We have natural ${\mathcal{A}}^{\unicode[STIX]{x1D703}}(X^{j})$ - $\mathsf{C}_{\unicode[STIX]{x1D708}}({\mathcal{A}}^{\unicode[STIX]{x1D703}})(X^{j}\,\cap \,X_{i}^{0})$ -bimodule structures on ${\mathcal{A}}^{\unicode[STIX]{x1D703}}(X^{j})/{\mathcal{A}}^{\unicode[STIX]{x1D703}}(X^{j}){\mathcal{A}}^{\unicode[STIX]{x1D703}}(X^{j})^{{>}0}$ and glue the bimodules corresponding to different $j$ together along the intersections $X^{i}\,\cap \,X^{j}$ (we have homomorphisms ${\mathcal{A}}^{\unicode[STIX]{x1D703}}(X^{i})\rightarrow {\mathcal{A}}^{\unicode[STIX]{x1D703}}(X^{i}\,\cap \,X^{j})$ that give rise to ${\mathcal{A}}^{\unicode[STIX]{x1D703}}(X^{i})/{\mathcal{A}}^{\unicode[STIX]{x1D703}}(X^{i}){\mathcal{A}}^{\unicode[STIX]{x1D703}}(X^{i})^{{>}0}\rightarrow {\mathcal{A}}^{\unicode[STIX]{x1D703}}(X^{i}\,\cap \,X^{j})/{\mathcal{A}}^{\unicode[STIX]{x1D703}}(X^{i}\,\cap \,X^{j}){\mathcal{A}}^{\unicode[STIX]{x1D703}}(X^{i}\,\cap \,X^{j})^{{>}0}$ and to $\mathsf{C}_{\unicode[STIX]{x1D708}}({\mathcal{A}}^{\unicode[STIX]{x1D703}}(X^{i}))\rightarrow \mathsf{C}_{\unicode[STIX]{x1D708}}({\mathcal{A}}^{\unicode[STIX]{x1D703}}(X^{i}\,\cap \,X^{j}))$ ). Similarly to the proof of (2) in Proposition 5.2, we get a sheaf of ${\mathcal{A}}^{\unicode[STIX]{x1D703}}$ - $\mathsf{C}_{\unicode[STIX]{x1D708}}({\mathcal{A}}^{\unicode[STIX]{x1D703}})$ -bimodules on $Y_{i}$ that we denote by ${\mathcal{A}}^{\unicode[STIX]{x1D703}}/{\mathcal{A}}^{\unicode[STIX]{x1D703}}{\mathcal{A}}^{\unicode[STIX]{x1D703},>0}$ . The following is a direct consequence of the construction.

Now we want to interpret $Y$ as a Lagrangian subvariety in $X\times X^{0}$ (we still use $X$ as in the paragraph preceding Lemma 5.6 and so can write $Y$ instead of $Y_{i}$ and $X^{0}$ instead of $X_{i}^{0}$ ). Namely, let $\unicode[STIX]{x1D704}$ denote the inclusion $Y{\hookrightarrow}X$ and $\unicode[STIX]{x1D70B}$ be the projection $Y\rightarrow X^{0}$ . We embed $Y$ into $X\times X^{0}$ via $(\unicode[STIX]{x1D704},\unicode[STIX]{x1D70B})$ . We equip $X\times X^{0}$ with the symplectic form $(\unicode[STIX]{x1D714},-\unicode[STIX]{x1D714}^{0})$ , where $\unicode[STIX]{x1D714}^{0}$ is the restriction of $\unicode[STIX]{x1D714}$ to  $X^{0}$ . With respect to this symplectic form, $Y$ is a Lagrangian subvariety. Further, ${\mathcal{A}}^{\unicode[STIX]{x1D703}}\,\widehat{\otimes }\,\mathsf{C}_{\unicode[STIX]{x1D708}}({\mathcal{A}}^{\unicode[STIX]{x1D703}})^{\text{opp}}$ is a quantization of $X\times X^{0}$ with period $(\unicode[STIX]{x1D706},-\unicode[STIX]{x1D706}^{0})$ , where $\unicode[STIX]{x1D706},\unicode[STIX]{x1D706}^{0}$ are periods of ${\mathcal{A}}^{\unicode[STIX]{x1D703}},\mathsf{C}_{\unicode[STIX]{x1D708}}({\mathcal{A}}^{\unicode[STIX]{x1D703}})$ .

5.5 Parabolic induction functor

Let $X$ be a conical symplectic resolution of $X_{0}$ . We assume that $X$ comes with a Hamiltonian action of a torus $T$ . Let $\mathfrak{C}$ stand for $\operatorname{Hom}(\mathbb{C}^{\times },T)$ . We introduce a pre-order $\prec ^{\unicode[STIX]{x1D706}}$ on $\mathfrak{C}$ as follows: $\unicode[STIX]{x1D708}\prec ^{\unicode[STIX]{x1D706}}\unicode[STIX]{x1D708}^{\prime }$ if:

1. (i) ${\mathcal{A}}_{\unicode[STIX]{x1D706}}({\mathcal{A}}_{\unicode[STIX]{x1D706}}^{{>}0,\unicode[STIX]{x1D708}}+({\mathcal{A}}_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D708}(\mathbb{C}^{\times })})^{{>}0,\unicode[STIX]{x1D708}^{\prime }})={\mathcal{A}}_{\unicode[STIX]{x1D706}}{\mathcal{A}}_{\unicode[STIX]{x1D706}}^{{>}0,\unicode[STIX]{x1D708}^{\prime }}$ ; and

2. (ii) the natural action of $\unicode[STIX]{x1D708}(\mathbb{C}^{\times })$ on $\mathsf{C}_{\unicode[STIX]{x1D708}^{\prime }}({\mathcal{A}}_{\unicode[STIX]{x1D706}})$ is trivial.

The pre-order extends naturally to $\mathfrak{C}_{\mathbb{Q}}:=\mathbb{Q}\,\otimes _{\mathbb{Z}}\mathfrak{C}$ .

The following lemma explains why this ordering is important.

The lemma shows that the Verma module functor can be studied in stages. This is what we mean by the parabolic induction.

Here is an example of two one-parameter subgroups that are comparable with respect to $\prec ^{\unicode[STIX]{x1D706}}$ . Pick one-parameter subgroups $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}:\mathbb{C}^{\times }\rightarrow T$ .

Proof. Let us write ${\mathcal{A}}$ for ${\mathcal{A}}_{\unicode[STIX]{x1D706}}$ . Let us prove the inclusion ${\mathcal{A}}^{{>}0,\unicode[STIX]{x1D708}}\subset {\mathcal{A}}{\mathcal{A}}^{{>}0,\unicode[STIX]{x1D708}^{\prime }}$ that will imply that ${\mathcal{A}}({\mathcal{A}}^{{>}0,\unicode[STIX]{x1D708}}+({\mathcal{A}}^{\unicode[STIX]{x1D708}(\mathbb{C}^{\times })})^{{>}0,\unicode[STIX]{x1D708}^{\prime }})\subset {\mathcal{A}}{\mathcal{A}}^{{>}0,\unicode[STIX]{x1D708}^{\prime }}$ .

The algebra $\operatorname{gr}({\mathcal{A}}^{{\geqslant}0,\unicode[STIX]{x1D708}})=\mathbb{C}[X_{0}]^{{\geqslant}0,\unicode[STIX]{x1D708}}$ is finitely generated, as in the proof of [Reference Gordon and LosevGL14, Lemma 3.1.2]. So, we can choose finitely many $T$ -semi-invariant generators of the ideal $\mathbb{C}[X_{0}]^{{>}0,\unicode[STIX]{x1D708}}$ in $\mathbb{C}[X_{0}]^{{\geqslant}0,\unicode[STIX]{x1D708}}$ , say $f_{1},\ldots ,f_{k}$ . Let $\tilde{f}_{1},\ldots ,\tilde{f}_{k}$ denote their lifts to $T$ -semi-invariant elements in ${\mathcal{A}}$ ; these lifts are generators of the ideal ${\mathcal{A}}^{{>}0,\unicode[STIX]{x1D708}}$ in ${\mathcal{A}}^{{\geqslant}0,\unicode[STIX]{x1D708}}$ . Let $a_{1},\ldots ,a_{k}>0$ be their weights for $\unicode[STIX]{x1D708}$ and $b_{1},\ldots ,b_{k}$ be their weights for $\unicode[STIX]{x1D705}$ . Take $m\in \mathbb{Z}^{{>}0}$ such that $ma_{i}+b_{i}>0$ for all $i$ . The elements $\tilde{f}_{1},\ldots ,\tilde{f}_{k}$ then lie in ${\mathcal{A}}^{{>}0,\unicode[STIX]{x1D708}^{\prime }}$ and so ${\mathcal{A}}{\mathcal{A}}^{{>}0,\unicode[STIX]{x1D708}}\subset {\mathcal{A}}{\mathcal{A}}^{{>}0,\unicode[STIX]{x1D708}^{\prime }}$ .

The inclusion ${\mathcal{A}}^{{>}0,\unicode[STIX]{x1D708}^{\prime }}\subset {\mathcal{A}}({\mathcal{A}}^{{>}0,\unicode[STIX]{x1D708}}+({\mathcal{A}}^{\unicode[STIX]{x1D708}(\mathbb{C}^{\times })})^{{>}0,\unicode[STIX]{x1D708}^{\prime }})$ is proved in a similar fashion. So, we get the equality in (i).

Let us check that $\unicode[STIX]{x1D708}(\mathbb{C}^{\times })$ acts trivially on $\mathsf{C}_{\unicode[STIX]{x1D708}^{\prime }}({\mathcal{A}})$ . We get

$$\begin{eqnarray}\mathsf{C}_{\unicode[STIX]{x1D708}^{\prime }}({\mathcal{A}})=\big({\mathcal{A}}/{\mathcal{A}}({\mathcal{A}}^{{>}0,\unicode[STIX]{x1D708}}\oplus ({\mathcal{A}}^{\unicode[STIX]{x1D708}(\mathbb{C}^{\times })})^{{>}0,\unicode[STIX]{x1D705}})\big)^{\unicode[STIX]{x1D708}^{\prime }(\mathbb{C}^{\times })}.\end{eqnarray}$$

Note that the expression in brackets is independent of $m$ and the weights of $\unicode[STIX]{x1D708}$ are all non-negative. From here it is easy to deduce that the invariants with respect to $\unicode[STIX]{x1D708}^{\prime }(\mathbb{C}^{\times })$ coincide with the invariants for $\unicode[STIX]{x1D708}(\mathbb{C}^{\times })\unicode[STIX]{x1D705}(\mathbb{C}^{\times })$ for $m$ large enough.◻

6.1 Main result

Suppose that we have a Hamiltonian action of a torus $T$ on $X$ (commuting with the contracting action) such that the fixed point set $X^{T}$ is finite. Let $\unicode[STIX]{x1D712}_{1},\ldots ,\unicode[STIX]{x1D712}_{N}$ be the characters of the $T$ -action on

$$\begin{eqnarray}\bigoplus _{p\in X^{T}}T_{p}X.\end{eqnarray}$$

The kernels $\ker \unicode[STIX]{x1D712}_{i},i=1,\ldots ,N,$ partition the cocharacter space $\operatorname{Hom}(\mathbb{C}^{\times },T)\,\otimes _{\mathbb{Z}}\mathbb{R}$ into polyhedral cones to be called chambers. If $\unicode[STIX]{x1D708}$ lies in the interior of a chamber (i.e., $\langle \unicode[STIX]{x1D712}_{i},\unicode[STIX]{x1D708}\rangle \neq 0$ for all $i$ ), then $X^{\unicode[STIX]{x1D708}(\mathbb{C}^{\times })}=X^{T}$ . Such one-parameter subgroups will be called generic. Otherwise, $X^{\unicode[STIX]{x1D708}(\mathbb{C}^{\times })}$ is infinite.

Let $\unicode[STIX]{x1D708}$ be a generic one-parameter subgroup of $T$ and $\unicode[STIX]{x1D708}_{0}$ be a one-parameter subgroup lying in the closure of the chamber containing $\unicode[STIX]{x1D708}$ . In this case we will write $\unicode[STIX]{x1D708}{\rightsquigarrow}\unicode[STIX]{x1D708}_{0}$ . Our goal is to produce a standardly stratified structure on ${\mathcal{O}}_{\unicode[STIX]{x1D708}}({\mathcal{A}}_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D703}})$ corresponding to $\unicode[STIX]{x1D708}_{0}$ . The first step is to define a pre-order on $X^{T}$ from $\unicode[STIX]{x1D708}_{0}$ .

Let $Z_{1},\ldots ,Z_{k}$ be the irreducible ( $=$ connected) components of $X^{\unicode[STIX]{x1D708}_{0}(\mathbb{C}^{\times })}$ . We define a partial order $\preccurlyeq _{\unicode[STIX]{x1D708}_{0}}$ on the set $\{Z_{1},\ldots ,Z_{k}\}$ as the transitive closure of the relation $Z_{i}\preccurlyeq _{\unicode[STIX]{x1D708}_{0}}Z_{j}$ specified by $Z_{i}\,\cap \,\overline{Y_{j}}\neq \varnothing$ , where $Y_{j}$ is the contracting locus of $Z_{j}$ , i.e.,

$$\begin{eqnarray}Y_{j}=\Big\{x\in X|\lim _{t\rightarrow 0}\unicode[STIX]{x1D708}_{0}(t)x\in Z_{j}\Big\}.\end{eqnarray}$$

For $p,p^{\prime }\in X^{T}$ , we set $p\preccurlyeq _{\unicode[STIX]{x1D708}_{0}}p^{\prime }$ if $p\in Z_{i},p^{\prime }\in Z_{j}$ with $Z_{i}\prec _{\unicode[STIX]{x1D708}_{0}}Z_{j}$ .