On inductive construction of Procesi bundles

TL;DR

This paper provides a geometric and representation-theoretic proof of an inductive formula for Procesi bundles on Hilbert schemes, leading to new partial results related to Haiman's $n!$ theorem.

Contribution

It introduces a new proof of the inductive formula for Procesi bundles and establishes weaker versions of the $n!$ theorem regarding the properties of the isospectral Hilbert scheme.

Findings

Proved the inductive formula for Procesi bundles using geometric methods.

Showed the normalization of Haiman's isospectral Hilbert scheme is Cohen-Macaulay and Gorenstein.

Established the normalization morphism is bijective, improving previous results.

Abstract

A Procesi bundle, a rank $n!$ vector bundle on the Hilbert scheme $H_n$ of $n$ points in $\mathbb{C}^2$, was first constructed by Mark Haiman in his proof of the $n!$ theorem by using a complicated combinatorial argument. Since then alternative constructions of this bundle were given by Bezrukavnikov-Kaledin and by Ginzburg. In this paper we give a geometric/ representation-theoretic proof of the inductive formula for the Procesi bundle that plays an important role in Haiman's construction. Then we use the inductive formula to prove a weaker version of the $n!$ theorem: the normalization of Haiman's isospectral Hilbert scheme is Cohen-Macaulay and Gorenstein, and the normalization morphism is bijective. This improves an earlier result of Ginzburg.

Equations

$${\hat{\alpha }}_{\ast }{\mathcal{O}}_{I{H}_{n,n-1}}={\mathcal{O}}_{I{H}_n}.$$

$$R{\mathrm{Hom}}_{{\mathcal{O}}_{H_n}}({\mathcal{P}}_n^H,\bullet ):D^b(\mathrm{Coh}{H}_n)\stackrel{\sim }{\to }{D}^b(\mathbb{C}[\underline{x},\underline{y}]{-mod}^{S_n}),$$

$$R\mathrm{End}({\mathcal{P}}_n^H)=\mathbb{C}[\underline{x},\underline{y}]\#S_n.$$

$${\alpha }_{\ast }({\beta }^{\ast }\left({\mathcal{P}}_{n-1}^H⊠{\mathcal{O}}_{{\mathbb{C}}^2}\right))={\mathcal{P}}_n^H.$$

$${\alpha }_{\ast }({\beta }^{\ast }\left({\mathcal{P}}_{n-1}^{BK}⊠{\mathcal{O}}_{{\mathbb{C}}^2}\right))={\mathcal{P}}_n^{BK},$$

$$c_1({\beta }_n^{\ast }{\mathcal{O}}_{n-1,♢}(1)),c_1({\alpha }_n^{\ast }{\mathcal{O}}_n(1)).$$

$${\mathcal{O}}_{n,n-1}(k,l):={\beta }_n^{\ast }({\mathcal{O}}_{n-1,♢}(k))\otimes {\alpha }_n^{\ast }{\mathcal{O}}_n(\ell ).$$

$${\mathcal{T}}_n{|}_{{\rho }_n^{-1}((V_n/S_n{)}^{{\wedge }_b})}\cong {\mathcal{O}}_{{\rho }_n^{-1}((V_n/S_n{)}^{{\wedge }_b})}^{\oplus n-1}\oplus {\mathcal{O}}_{{\rho }_n^{-1}((V_n/S_n{)}^{{\wedge }_b})}(1).$$

$${\tau }_n:{\tau }_n^{-1}({\rho }_n^{-1}((V_n/S_n{)}^{{\wedge }_b}))\to {\rho }^{-1}((V_n/S_n{)}^{{\wedge }_b}.$$

$$\mathsf{Per}({\mathcal{D}}^{opp})=-\mathsf{Per}(\mathcal{D}).$$

$$\mathcal{D}:=({\phi }_{\ast }\left([D_R/D_R\Phi (\mathfrak{g})]{|}_{(T^{\ast }R{)}^{\theta -s}}\right){)}^G.$$

$$\mathsf{Per}(\mathcal{D})=-\frac{1}{2}.$$

$${\underline{\mathcal{D}}}^{\mathbb{F}}:=({\phi }_{\ast }^{(1)}\left([D(R^{\mathbb{F}}){|}_{({\mu }^{(1)}{)}^{-1}(0{)}^{\theta -s}}{]}^{\mathfrak{g}}\right){)}^{G^{\mathbb{F}(1)}}.$$

$${\tilde{A}}_n^{\mathbb{F}\wedge }={\hat{\mathcal{B}}}_n^{\mathbb{F}}{\otimes }_{{\mathcal{A}}_n^{\mathbb{F}\wedge }}{\hat{\mathcal{B}}}_n^{o,\mathbb{F}}.$$

$${\hat{\mathcal{P}}}_n^{\mathbb{F}}:={\hat{\mathcal{B}}}_n^{\mathbb{F}}{\otimes }_{{\mathcal{A}}_n^{\mathbb{F}\wedge }}{\hat{\mathcal{E}}}^{\mathbb{F}}.$$

$$\mathrm{End}({\hat{\mathcal{P}}}_n^{\mathbb{F}})={\hat{\mathcal{B}}}_n^{\mathbb{F}}{\otimes }_{{\mathcal{A}}_n^{\mathbb{F}\wedge }}\mathrm{End}({\hat{\mathcal{E}}}_{\mathbb{F}}){\otimes }_{{\mathcal{A}}_n^{\mathbb{F}\wedge }}{\hat{\mathcal{B}}}_n^{o,\mathbb{F}}={\tilde{A}}_n^{\mathbb{F}\wedge }.$$

$${\mathrm{Coh}}^{T_c^{(1)}}(H_n^{\mathbb{F}(1)})\stackrel{\sim }{\to }{\mathrm{Coh}}^{T_c^{(1)}}(H_n^{\mathbb{F}(1)\wedge }).$$

$${\iota }^{\ast }(\mathsf{Per}({\mathcal{D}}_{\hslash }))=-\frac{1}{2}{c}_1(K_Y).$$

$${\iota }^{\ast }(\mathsf{Per}({\mathcal{D}}_n^{opp}\hat{\otimes }{\mathcal{D}}_{n-1,♢}))=-\frac{1}{2}{c}_1(K_Y),$$

$$\left({\mathcal{D}}_{n-1,♢}\hat{\otimes }{\mathcal{D}}_n^{opp}\right){\otimes }_{({\mathcal{A}}_{n-1,♢}\otimes {\mathcal{A}}_n^{opp})}{\mathcal{A}}_{n-1,♢}\stackrel{\sim }{\to }{\mathcal{D}}_{n,n-1}.$$

$$\mathrm{Loc}(\bullet ):=\left({\mathcal{D}}_{n-1,♢}\hat{\otimes }{\mathcal{D}}_n^{opp}\right){\otimes }_{({\mathcal{A}}_{n-1,♢}\otimes {\mathcal{A}}_n^{opp})}(\bullet )$$

$${\mathcal{D}}_{n,n-1}^{\prime \mathsf{R}}:=({\mathcal{D}}_n^{\mathsf{R},opp}\hat{\otimes }{\mathcal{D}}_{n-1,♢}^{\mathsf{R}}){\otimes }_{{\mathcal{A}}_n^{\mathsf{R},opp}\otimes {\mathcal{A}}_{n-1,♢}^{\mathsf{R}}}{\mathcal{A}}_{n-1,♢}^{\mathsf{R}}.$$

$${\mathrm{Coh}}^{T_c}(H_n^{\mathbb{F}(1)}\times H_{n-1,♢}^{\mathbb{F}(1)}\times {\mathbb{A}}^1)\stackrel{\sim }{\to }{\mathrm{Coh}}^{T_c}({\mathcal{O}}_{H_n^{\mathbb{F}(1)}\times H_{n-1,♢}^{\mathbb{F}(1)}}[[\hslash ]]).$$

$${\mathrm{Fr}}_{\ast }{\mathcal{D}}_{n,n-1}^{\mathbb{F}}\subset {\iota }_{\ast }{\iota }^{\ast }{\mathrm{Fr}}_{\ast }{\mathcal{D}}_{n,n-1}^{\mathbb{F}}.$$

$$\left({\underline{\mathcal{D}}}_{n-1,♢}^{\mathbb{F}}\otimes {\underline{\mathcal{D}}}_n^{\mathbb{F},opp})\right){|}_{H_{n,n-1}^{\mathbb{F}(1)}}.$$

$${\alpha }_{n\ast }{\beta }_n^{\ast }{\mathcal{P}}_{n-1,♢}^{\mathbb{F}}(k)\cong {\mathcal{P}}_n^{\mathbb{F}}(\ell ).$$

$$D^b({\mathcal{A}}_{n-1,♢}^{\mathbb{F}opp}-mod)\to D^b(\mathrm{Coh}({\mathcal{D}}_{n-1,♢}^{\mathbb{F},opp}))\to D^b(\mathrm{Coh}({\mathcal{D}}_n^{\mathbb{F},opp})\to D^b({\mathcal{A}}_n^{\mathbb{F}opp}-mod)$$

$${\underline{\mathcal{D}}}_{n,n-1}^{\mathbb{F}\wedge }\cong {\alpha }_n^{(1)\ast }\left({\hat{\mathcal{E}}}_n^{\mathbb{F}\ast }\right)\otimes {\beta }_n^{(1)\ast }({\hat{\mathcal{E}}}_{n-1,♢}^{\mathbb{F}})\otimes \mathcal{L}.$$

$${\mathcal{D}}_{n,n-1}^{\mathbb{F}\wedge }\cong {\alpha }_n^{(1)\ast }({\hat{\mathcal{E}}}_n(k{)}^{\mathbb{F}\ast })\otimes {\beta }_n^{(1)\ast }({\hat{\mathcal{E}}}_{n-1,♢}(\ell {)}^{\mathbb{F}}).$$

$${\hat{\mathcal{B}}}_n^{\mathbb{F}}{\otimes }_{{\mathcal{A}}_n^{\mathbb{F}\wedge }}{\hat{\mathcal{B}}}_n^{o,\mathbb{F}}={\tilde{A}}_n^{\mathbb{F}(1)\wedge },{\hat{\mathcal{B}}}_{n-1,♢}^{\mathbb{F}}{\otimes }_{{\mathcal{A}}_{n-1,♢}^{\mathbb{F}\wedge }}{\hat{\mathcal{B}}}_{n-1,♢}^{o,\mathbb{F}}={\tilde{A}}_{n-1,♢}^{\mathbb{F}(1)\wedge },$$

Full text

MSC 2010: 14E16, 53D55, 16G99

1. Introduction

1.1. $n!$ theorem

The Hilbert scheme $H_{n}:=\operatorname{Hilb}_{n}(\mathbb{C}^{2})$ parameterizes the codimension $n$ ideals in $\mathbb{C}[x,y]$. It is known to be a smooth irreducible algebraic variety of dimension $2n$. It comes with the Hilbert-Chow morphism $\rho_{n}:H_{n}\rightarrow V_{n}/S_{n}$, where we write $V_{n}$ for $(\mathbb{C}^{2})^{\oplus n}$ and the symmetric group $S_{n}$ acts by permuting the $n$ copies of $\mathbb{C}^{2}$. The morphism sends a point in $H_{n}$ – a codimension $n$ ideal in $\mathbb{C}[x,y]$ – to its support counted with multiplicities. This morphism is known to be a resolution of singularities.

Following Haiman, we define the isospectral Hilbert scheme $IH_{n}$ as the fiber product $V_{n}\times_{V_{n}/S_{n}}H_{n}$ with reduced scheme structure. By definition, there is a finite morphism $\eta:IH_{n}\rightarrow H_{n}$. It is the quotient morphism for the natural action of $S_{n}$ on $IH_{n}$.

Here is a geometric version Haiman’s $n!$ theorem, [H1, Theorem 3.1], (there is also an elementary version that has to do with the spaces of partial derivatives of certain two-variable generalizations of the Vandermonde determinant, that version follows from the geometric one).

Theorem 1.1**.**

The variety $IH_{n}$ is Cohen-Macaulay and Gorenstein.

Note that, in order for $IH_{n}$ to be Cohen-Macaulay, we do need to consider $V_{n}\times_{V_{n}/S_{n}}H_{n}$ with its reduced scheme structure: the fiber product with the natural scheme structure can be shown to be non-reduced while generically reduced. This can never happen for Cohen-Macaulay schemes.

Let us explain a motivation behind Theorem 1.1. Note that the two-dimensional torus $T:=(\mathbb{C}^{\times})^{2}$ acts on $\mathbb{C}^{2}$ in a natural way: $(t_{1},t_{2}).(x,y)=(t_{1}^{-1}x,t_{2}^{-1}y)$. The action induces an action on $V_{n}$, on $\mathbb{C}[x,y]$ by algebra automorphisms and hence an action on $H_{n}$. The Hilbert-Chow map $H_{n}\rightarrow V_{n}/S_{n}$ is $T$-equivariant so we get a $T$-action on $IH_{n}$ as well. The $T$-fixed points in $H_{n}$ precisely correspond to the monomial codimension $n$ ideals in $\mathbb{C}[x,y]$. The latter are naturally labelled by Young diagrams with $n$ boxes: the diagram corresponding to a monomial ideal indicates which monomials do not lie in the ideal. Let us write $x_{\lambda}$ for the fixed point labelled by the Young diagram $\lambda$.

Now consider the sheaf $\mathcal{P}^{H}_{n}:=\eta_{*}\mathcal{O}_{IH_{n}}$ (the superscript “H” is for Haiman). Since $IH_{n}$ is Cohen-Macaulay and $\eta$ is finite, $\eta$ is flat. The degree is easily seen to be $n!$. So $\mathcal{P}^{H}_{n}$ is a degree $n!$ vector bundle. The idea to look for such a bundle was due to Procesi, so Haiman called $\mathcal{P}^{H}_{n}$ the Procesi bundle. The bundle $\mathcal{P}^{H}_{n}$ is $T$-equivariant. In particular, its fibers at the fixed points, $\mathcal{P}^{H}_{\lambda}:=(\mathcal{P}^{H}_{n})_{x_{\lambda}}$ are bigraded $S_{n}$-modules. Another hard theorem from [H1] says that the Frobenius character of the bigraded $S_{n}$-module $(\mathcal{P}^{H}_{n})_{x_{\lambda}}$ is the modified Macdonald polynomial $\tilde{H}_{\lambda}(q,t)$. This, in particular, proves the famous Macdonald positivity conjecture that says that the Macdonald polynomials are Schur positive.

1.2. Inductive construction

Let us say a few words about Haiman’s proof of Theorem 1.1, see [H1, Sections 3,4] and also [H3, Sections 5.6] for an overview. The construction is inductive in nature and utilizes the nested Hilbert scheme $H_{n,n-1}$. This scheme parameterizes pairs of ideals $J\subset J^{\prime}$ in $\mathbb{C}[x,y]$ such that $\operatorname{codim}_{\mathbb{C}[x,y]}J=n$ and $\operatorname{codim}_{\mathbb{C}[x,y]}J^{\prime}=n-1$. It turns out, [C], that $H_{n,n-1}$ is an irreducible smooth variety of dimension $2n$. It comes with two morphisms $\beta:H_{n,n-1}\rightarrow H_{n-1}\times\mathbb{C}^{2}$ sending $(J^{\prime}\subset J)$ to $(J^{\prime},\operatorname{Supp}(J^{\prime}/J))$ and $\alpha:H_{n,n-1}\rightarrow H_{n}$ forgetting $J^{\prime}$.

Set $IH_{n,n-1}:=H_{n,n-1}\times_{H_{n-1}}IH_{n-1}$. This scheme comes with a natural morphism to $IH_{n}$, denote it by $\hat{\alpha}$, it is induced by $\alpha$. Haiman deduced Theorem 1.1 from the following equality

[TABLE]

Here and below we will use the notation like $\hat{\alpha}_{*}$ for the derived functor.

The proof of (1.1) is based on Haiman’s polygraph theorem, [H1, Theorem 4.1], which is proved in [H1] by explicit combinatorial/ commutative-algebraic computations. Since these computations are extremely complicated, there were several attempts to under various parts of Haiman’s construction more conceptually. The present paper contributes to this goal.

1.3. Related developments

Before explaining results of this paper we want to discuss some subsequent developments related to Haiman’s work starting with a construction that ours is based upon: a construction of the Procesi bundle due to Bezrukavnikov and Kaledin, [BK2].

An important property of the Procesi bundle $\mathcal{P}^{H}_{n}$ observed by Haiman in [H2] is the derived McKay equivalence

[TABLE]

where in the target we write $\underline{x}$ for $x_{1},\ldots,x_{n}$, $\underline{y}$ for $y_{1},\ldots,y_{n}$, and the superscript “$S_{n}$” means that we consider the category of $S_{n}$-equivariant modules. This equivalence sends $\mathcal{P}^{H}_{n}$ to the smash-product algebra $\mathbb{C}[\underline{x},\underline{y}]\#S_{n}$ (whose category of modules is $\mathbb{C}[\underline{x},\underline{y}]\operatorname{-mod}^{S_{n}}$). A consequence (and, in fact, an equivalent condition thanks to [BK2, Proposition 2.2]) is that

[TABLE]

On the other hand, derived equivalences of this sort are classical in geometric Representation theory, where they come from the derived Beilinson-Bernstein type localization theorem. Inspired by the version of this theorem in positive characteristic proved in [BMR], in [BK2] Bezrukavnikov and Kaledin established a derived McKay equivalence and constructed an analog of a Procesi bundle on any symplectic resolution of any symplectic quotient singularity, a basic example is the resolution $H_{n}$ of $V_{n}/S_{n}$. The latter variety admits a standard quantization, $\mathsf{W}(V_{n})^{S_{n}}$, where we write $\mathsf{W}(V_{n})$ for the Weyl algebra of the symplectic vector space $V_{n}$. Over an algebraically closed field $\mathbb{F}$ of positive characteristic, the algebra $\mathsf{W}(V_{n})$ is an Azumaya algebra over the Frobenius twist $V_{n}^{(1)}$. The restrictions of $\mathsf{W}(V_{n})^{S_{n}}$ and $\mathbb{F}[V_{n}^{(1)}]\#S_{n}$ to the formal neighborghood of [math] in $V_{n}^{(1)}$ are Morita equivalent provided the characteristic is large enough. Further, one can find a “Frobenius constant” quantization $\underline{\mathcal{D}}$ of $H_{n}$ whose algebra of (derived) global sections (automatically a quantization of $V_{n}/S_{n}$) is $\mathsf{W}(V_{n})^{S_{n}}$. Being Frobenius constant means, in particular, that $\underline{\mathcal{D}}$ can be viewed as an Azumaya algebra on $H_{n}^{(1)}$. The restriction of $\underline{\mathcal{D}}$ to the formal neighborhood of the zero fiber of $H_{n}^{(1)}\rightarrow V_{n}^{(1)}/S_{n}$ splits. From the indecomposable summands of a splitting bundle one can form an analog of a Procesi bundle. Then thanks to the rigidity (the absence of higher self-extensions) one can first extend this analog of a Procesi bundle to the whole variety $H_{n}^{(1)}$ and then lift to characteristic [math]. One gets a vector bundle $\mathcal{P}_{n}^{BK}$ on $H_{n}$ satisfying (1.2).

The bundle $\mathcal{P}_{n}^{BK}$ can be shown to coincide with $\mathcal{P}_{n}^{H}$ thanks to results of [L2], where “abstract” Procesi bundles were classified: it was shown that there are exactly two abstract normalized Procesi bundles that are dual to one another. In this paper we reprove Theorem 1.1 using the bundle $\mathcal{P}_{n}^{BK}$. Let us also point out that the Macdonald positivity was reproved (and generalized) in [BF] by using the construction of $\mathcal{P}_{n}^{BK}$ from [BK2].

We would also like to mention several other related developments, although they do not play any role in our proofs. Ginzburg proved that the normalization of $IH_{n}$ is Cohen-Macaulay and Gorenstein using the Hodge filtration on the Hotta-Kashiwara D-module, see [Gi, Proposition 8.2.4]. The variety $IH_{n}$ was shown to be normal by Haiman but presently there is no independent proof of the normality. Gordon, [Go], used Ginzburg’s construction to deduce the Macdonald positivity. There are also other proofs of the Macdonald positivity, e.g., [GH].

1.4. Results and ideas of proof

We now proceed to explaining the ideas of our proof. First, let us record a consequence of (1.1):

[TABLE]

Our goal is to establish an analog of this equality for the Procesi bundle constructed by Bezrukavnikov and Kaledin:

Theorem 1.2**.**

We have a $\mathbb{C}[\underline{x},\underline{y}]\#S_{n-1}$-linear isomorphism

[TABLE]

We will use this theorem to strengthen a result of Ginzburg, [Gi, Proposition 8.2.4].

Theorem 1.3**.**

The normalization of $IH_{n}$ is Cohen-Macaulay and Gorenstein. Moreover the normalization morphism for $IH_{n}$ is bijective.

In the remainder of this section we will discuss the ideas of our proof of (1.4).

We will prove a “quantum analog” of (1.4) in characteristic $p\gg 0$. Namely, note that $H_{n,n-1}$ is a lagrangian subvariety in $H_{n}\times(H_{n-1}\times\mathbb{C}^{2})$. We will show that, over $\mathbb{C}$, the sheaf $\mathcal{O}_{H_{n,n-1}}$ can be quantized to a filtered $\mathcal{D}_{n-1,\diamondsuit}$-$\mathcal{D}_{n}$-bimodule $\mathcal{D}_{n,n-1}$ (where $\mathcal{D}_{n}$ is the characteristic [math] version of the quantization of $H_{n}$ used in [BK2] to produce the Procesi bundle and $\mathcal{D}_{n-1,\diamondsuit}$ is an analogous quantization of $H_{n-1}\times\mathbb{C}^{2}$). We will show that $\Gamma(\mathcal{D}_{n,n-1})=\mathsf{W}(V_{n})^{S_{n-1}}$, while the higher cohomologies of $\mathcal{D}_{n,n-1}$ vanish.

Then we will show that the quantization $\mathcal{D}_{n,n-1}$ can be reduced mod $p$ for $p\gg 0$. Let us denote corresponding quantization by $\mathcal{D}^{\mathbb{F}}_{n,n-1}$. We still have $R\Gamma(\mathcal{D}_{n,n-1}^{\mathbb{F}})=\mathsf{W}(V_{n}^{\mathbb{F}})^{S_{n-1}}$. Also we will show that $\mathcal{D}^{\mathbb{F}}_{n,n-1}$ gives rise to a splitting bundle for the restriction to $H_{n,n-1}^{\mathbb{F},(1)}$ of the Azumaya algebra arising from $\mathcal{D}_{n}^{\mathbb{F},opp}\otimes\mathcal{D}_{n-1,\diamondsuit}^{\mathbb{F}}$. Using this splitting bundle and the Bezrukavnikov-Kaledin construction of the Procesi bundle (and some additional work) we deduce (1.4) over $\mathbb{F}$ from here. Using this, we will prove Theorems 1.2 and 1.3 over $\mathbb{C}$.

1.5. Conventions and notation

In this paper we work with various base fields and rings. They include $\mathbb{C}$, $\mathbb{F}:=\overline{\mathbb{F}}_{p}$ for $p$ sufficiently large, a large finite localization $\mathsf{R}$ of $\mathbb{Z}$ and some others. The base ring is indicated as a supercript, unless it is $\mathbb{C}$, in which case we skip the superscript $\mathbb{C}$.

All pullback and pushforward functors are derived, while $\Gamma$ denotes the usual non-derived global section functor.

Notation for varieties and algebras. Throughout the paper we use the following notation for varieties and (commutative and noncommutative) algebras:

• •

$V_{n}:=(\mathbb{C}^{2})^{\oplus n}$. By $x_{1},y_{1},\ldots,x_{n},y_{n}$ we denote the natural basis in $V_{n}$ so that $\mathbb{C}[V_{n}]=\mathbb{C}[x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}]$.

• •

$V_{n}^{0}$ is the regular locus in $V_{n}$ consisting of $n$-tuples of pairwise distinct points.

• •

$V_{n}^{1}$ is the open locus in $V_{n}$ consisting of all points whose stabilizer in $S_{n}$ is either trivial or is generated by a single transposition.

• •

$H_{n}$ is the Hilbert scheme of $n$ points in the plane.

• •

$H_{n,n-1}$ is the nested Hilbert scheme parameterizing inclusions $J^{\prime}\subset J$ with $J\in H_{n},J^{\prime}\in H_{n-1}$.

• •

$IH_{n}$ is the isospectral Hilbert scheme, i.e., $V_{n}\times_{V_{n}/S_{n}}H_{n}$ with reduced scheme structure.

• •

$U_{n}$ is the universal degree $n$ family over $H_{n}$.

• •

$\mathsf{W}(V_{n})$ is the Weyl algebra of the symplectic vector space $V_{n}$.

• •

$\mathcal{A}_{n}:=\mathsf{W}(V_{n})^{S_{n}}$.

• •

$\widetilde{A}_{n}:=\mathbb{C}[V_{n}]\#S_{n}$.

• •

$X_{n}$ is the relative spectrum of the Procesi bundle on $H_{n}$, constructed in Section 6.

• •

$X_{n,n-1}:=X_{n-1}\times_{H_{n-1}}H_{n,n-1}$.

We also often use the subscript $\diamondsuit$. The meaning is the multiplication by $\mathbb{C}^{2}$ for varieties or by $\mathbb{C}[V_{2}]$ or $\mathsf{W}(V_{2})$ for the algebras above. For example, $V^{0}_{n-1,\diamondsuit}$ stands for $V^{0}_{n-1}\times\mathbb{C}^{2}$ (where the second factor corresponds to the last two coordinates) and $\mathcal{A}_{n-1,\diamondsuit}$ stands for $\mathcal{A}_{n-1}\otimes\mathsf{W}(V_{2})$.

As usual, the superscript “(1)” indicates the Frobenius twist.

Notation for morphisms:

• •

$\alpha_{n}$ is the natural morphism $H_{n,n-1}\rightarrow H_{n}$.

• •

$\bar{\alpha}_{n}$ is the natural morphism $H_{n,n-1}\rightarrow U_{n}$.

• •

$\widetilde{\alpha}_{n}$ is the natural morphism $X_{n,n-1}\rightarrow X_{n}$ defined in Section 6.

• •

$\beta_{n}$ is the natural morphism $H_{n,n-1}\rightarrow H_{n-1,\diamondsuit}$.

• •

$\widetilde{\beta}_{n}$ is the natural morphism $X_{n,n-1}\rightarrow X_{n-1,\diamondsuit}$.

• •

$\eta_{n}$ is the natural morphism $IH_{n}\rightarrow H_{n}$.

• •

$\mu$ stands for a moment map.

• •

$\pi_{n}$ is the natural morphism $X_{n}\rightarrow H_{n}$ introduced in Section 6. Similarly, $\pi_{n,n-1}:X_{n,n-1}\rightarrow H_{n,n-1}$ is the natural morphism.

• •

$\rho_{n}:H_{n}\rightarrow V_{n}/S_{n}$ is the Hilbert-Chow morphism. Similarly, we write $\widetilde{\rho}_{n},\widetilde{\rho}_{n,n-1},\bar{\rho}_{n}$ for the natural morphisms $X_{n}\rightarrow V_{n},X_{n,n-1}\rightarrow V_{n},U_{n}\rightarrow V_{n}/S_{n-1}$, respectively.

• •

$\tau_{n}$ is also the natural morphism $U_{n}\rightarrow H_{n}$.

Finally, let us list our notation for various sheaves.

• •

$\mathcal{D}_{n}$ denotes a certain microlocal quantization of $H_{n}$ recalled in Section 3.1.

• •

$\underline{\mathcal{D}}^{\mathbb{F}}_{n}$ denotes a Frobenius constant quantization of $H_{n}^{\mathbb{F}}$ obtained from $\mathcal{D}^{\mathbb{F}}_{n}$.

• •

$\mathcal{D}_{n,n-1}$ denotes a certain microlocal quantization of the lagrangian subvariety $H_{n,n-1}\subset H_{n}\times H_{n-1,\diamondsuit}$ constructed in Section 4, and $\underline{\mathcal{D}}_{n,n-1}^{\mathbb{F}}$ is the “Frobenius constant” version.

• •

$K_{Y}$ denotes the canonical bundle of a smooth variety $Y$.

• •

$\mathcal{O}_{n},\mathcal{O}_{n,n-1}$ denote the structure sheaves of $H_{n},H_{n,n-1}$. More generally, we write $\mathcal{O}_{n}(k)$ for the line bundle on $H_{n}$ corresponding to $k\in\mathbb{Z}$ and $\mathcal{O}_{n,n-1}(k,\ell)$ for $\beta_{n}^{*}\mathcal{O}_{n-1}(k)\otimes\alpha_{n}^{*}\mathcal{O}_{n}(\ell)$, a line bundle on $H_{n,n-1}$.

• •

$\mathcal{P}_{n}$ is the Procesi bundle on $H_{n}$ whose construction is recalled in Section 3.2.

• •

$\mathcal{T}_{n}:=\tau_{n*}\mathcal{O}_{U_{n}}$ is the tautological bundle on $H_{n}$.

Acknowledgements: I would like to thank Roman Bezrukavnikov, Pavel Etingof, Victor Ginzburg, Evgeny Gorsky, Andrei Negut, Alexei Oblomkov, and Lev Rozansky for stimulating discussions. I would also like to thank Roman Bezrukavnikov for the discussion that allowed me to find a fatal mistake in a previous version of this paper that claimed a proof of the $n!$ theorem. This work has been funded by the Russian Academic Excellence Project ’5-100’.

2. Preliminaries on Hilbert schemes

2.1. Hilbert scheme $H_{n}$

First of all, let us recall a construction of $H_{n}$ via the Hamiltonian reduction. Consider the space $R:=\operatorname{End}(\mathbb{C}^{n})\oplus\mathbb{C}^{n}$. This space has a natural action of $G:=\operatorname{GL}_{n}$.

Consider the cotangent bundle $T^{*}R=R\oplus R^{*}$. The group $G$ acts on $T^{*}R$ and the action is Hamiltonian. The moment map can be described as follows. First, we can identify $\operatorname{End}(\mathbb{C}^{n})^{*}$ with $\operatorname{End}(\mathbb{C}^{n})$ via the trace form. Then we can view an element of $T^{*}R$ as a quadruple $(A,B,i,j)$ for $A,B\in\operatorname{End}(\mathbb{C}^{n}),i\in\mathbb{C}^{n},j\in\mathbb{C}^{n*}$. The moment map $\mu:T^{*}R\rightarrow\mathfrak{g}$ is given by $\mu(A,B,i,j)=AB+ij$. Now we are going to describe the pull-back map $\mu^{*}:\mathfrak{g}\rightarrow\mathbb{C}[T^{*}R]$. Consider the velocity vector field map $\mathfrak{g}\rightarrow\operatorname{Vect}(R),\xi\mapsto\xi_{R}$. We can view $\operatorname{Vect}(R)$ as a subspace of $\mathbb{C}[T^{*}R]=\mathbb{C}[R\oplus R^{*}]$ that consists of all functions that have degree $1$ in $R^{*}$. It is easy to show that $\mu^{*}(\xi)=\xi_{R}$.

Consider the character $\theta:=\det^{-1}$. By definition, the $\theta$-stable locus in $T^{*}R$ consists of the quadruples $(A,B,i,j)$ such that $\mathbb{C}\langle A,B\rangle i=\mathbb{C}^{n}$. This locus will be denoted by $(T^{*}R)^{\theta-s}$. Note that the action of $G$ on $(T^{*}R)^{\theta-s}$ is free. For $(A,B,i,j)\in\mu^{-1}(0)^{\theta-s}:=\mu^{-1}(0)\cap(T^{*}R)^{\theta-s}$ we necessarily have $j=0$ and hence $[A,B]=0$.

By definition, $H_{n}$ is the GIT Hamiltonian reduction of $T^{*}R$ by the action of $G$ with character $\theta$. Equivalently, $H_{n}$ is the GIT quotient $\mu^{-1}(0)^{\theta-s}/G$. Below we will write $\mathcal{O}_{n}$ for the structure sheaf $\mathcal{O}_{H_{n}}$ of $H_{n}$. We have an ample line bundle $\mathcal{O}_{n}(1)$ on $H_{n}$ given by $\varphi_{*}(\mathcal{O}_{\mu^{-1}(0)^{\theta-s}}\otimes\det)^{G}$, where we write $\varphi$ for the quotient morphism $\mu^{-1}(0)^{\theta-s}\rightarrow H_{n}$.

We have a natural morphism of quotients $\rho_{n}:H_{n}\rightarrow\mu^{-1}(0)/\!/G$. The target variety is identified with $V_{n}/S_{n}$ and the morphism is the Hilbert-Chow map that sends a codimension $n$ ideal to its support with multiplicities. Also recall that $H_{n}$ is a resolution of singularities of $V_{n}/S_{n}$. The morphism $\rho_{n}$ is an isomorphism over the smooth locus $(V_{n}/S_{n})^{reg}$. This locus coincides with $V_{n}^{0}/S_{n}$, where we write $V_{n}^{0}$ for the locus of $n$ different points in $\mathbb{C}^{2}$.

Example 2.1**.**

It is easy to see that $H_{2}=\operatorname{Bl}_{\Delta}(V_{2}/S_{2})$, the blow-up of the diagonal.

Also note that $H_{n}$ comes with several additional structures. For examples, as a Hamiltonian reduction of a smooth symplectic variety by a free $G$-action, $H_{n}$ carries a natural symplectic form, $\omega$. Since $H_{n}$ is a symplectic resolution of a normal variety, by the Grauert-Riemenschneider theorem, we have the following claim.

Lemma 2.2**.**

We have $H^{i}(H_{n},\mathcal{O}_{n})=0$ for $i>0$ and $\Gamma(\mathcal{O}_{n})=\mathbb{C}[V_{n}]^{S_{n}}$.

The variety $H_{n}$ also comes with an action of the two-dimensional torus $T$. It is induced from the $T$-action on $T^{*}R$ given by $(t_{1},t_{2})(r,\alpha)=(t_{1}^{-1}r,t_{2}^{-1}\alpha)$ for $r\in R,\alpha\in R^{*}$. For the induced action on $H_{n}$ we have $(t_{1},t_{2}).\omega=t_{1}t_{2}\omega$. In particular, the action of $T_{h}:=\{(t,t^{-1})\}\subset T$ preserves $\omega$ and, moreover, is Hamiltonian. The action of $T_{c}:=\{(t,t)\}\subset T$ is contracting and rescales the symplectic form $\omega$ by $t^{2}$. Below this action will be called contracting.

The $T$-fixed points correspond to the monomial ideals in $\mathbb{C}[x,y]$ and hence are labelled by the partitions of $n$: for a partition $\lambda$ of $n$, we write $x_{\lambda}$ for the corresponding fixed point.

Now let us recall how to compute the group $\operatorname{Pic}(H_{n})$ and the space $H^{2}(H_{n},\mathbb{C})$. Pick the subtorus of the form $\{(t,t^{N})\}\subset T$ for $N\gg 0$, it is contracting and has finitely many fixed points. The open cell for the Bialynicki-Birula decomposition corresponds to the fixed point $x_{(n)}$. And there is one codimension $1$ cell corresponding to $x_{(n-1,1)}$. It follows that $\operatorname{Pic}(H_{n})\cong\mathbb{Z}$, in fact, this group is generated by the line bundle $\mathcal{O}_{n}(1)$ constructed above. And $H^{2}(H_{n},\mathbb{C})\cong\mathbb{C}$ is generated by $c_{1}(\mathcal{O}_{n}(1))$. Note that the closure of the codimension 1 cell coincides with $H_{n}\setminus\rho_{n}^{-1}(V_{n}^{0}/S_{n})$.

Let us describe the structure of $H_{n}$ over a neighborhood of a generic point in $(V_{n}/S_{n})^{sing}$. The following lemma is classical.

Lemma 2.3**.**

Let $b\in V_{n}/S_{n}$ be an $n$-tuple with precisely one pair of repeated points and $(V_{n}/S_{n})^{\wedge_{b}}$ denote its formal neighborhood in $V_{n}/S_{n}$. Then $\rho_{n}^{-1}\left((V_{n}/S_{n})^{\wedge_{b}}\right)$ is isomorphic to the preimage of $(V_{2}/S_{2}\times V_{n-2})^{\wedge_{0}}$ in $\operatorname{Bl}_{\Delta}(V_{2}/S_{2})\times V_{n-2}$.

To finish the section, let us discuss possible rings of definition. The scheme $H_{n}$ is defined over $\mathbb{Z}$. For general reasons, the corresponding $\mathsf{R}$-form $H_{n}^{\mathsf{R}}$ is smooth and symplectic over a finite localization $\mathsf{R}$ of $\mathbb{Z}$. Further it is a resolution of singularities of $V_{n}^{\mathsf{R}}/S_{n}$. We will also change the base from $\mathsf{R}$ to $\mathbb{F}:=\overline{\mathbb{F}}_{p}$ for $p\gg 0$ and all results in the previous paragraphs of this section continue to hold.

2.2. Nested Hilbert scheme $H_{n,n-1}$

Recall the nested Hilbert scheme $H_{n,n-1}$ parameterizing pairs of ideals $J\subset J^{\prime}\subset\mathbb{C}[x,y]$ such that $\dim\mathbb{C}[x,y]/J=n,\dim\mathbb{C}[x,y]/J^{\prime}=n-1$. We have the following fundamental result on $H_{n,n-1}$, [C, Section 0.2] (the theorem was actually proved earlier by A. Tikhomirov in an unpublished preprint).

Proposition 2.4**.**

The scheme $H_{n,n-1}$ is smooth and irreducible of dimension $2n$.

Here and below to simplify the notation we will write $H_{n-1,\diamondsuit}$ for $H_{n-1}\times\mathbb{C}^{2}$. As was mentioned in the introduction, the variety $H_{n,n-1}$ comes with two morphisms: $\alpha_{n}:H_{n,n-1}$ that sends $(J\subset J^{\prime})$ to $J$ and $\beta_{n}:H_{n,n-1}\rightarrow H_{n-1,\diamondsuit}$, which sends $(J\subset J^{\prime})$ to the pair of $J^{\prime}$ and the support of $J^{\prime}/J$.

Consider the morphism $\rho_{n,n-1}:H_{n,n-1}\rightarrow V_{n}/S_{n-1}$ given by $\rho_{n-1}\circ\beta_{n}$. Note that it is an isomorphism over $V_{n}^{0}/S_{n-1}$.

Lemma 2.5**.**

The morphism $\rho_{n,n-1}$ gives rise to an isomorphism $\mathbb{C}[V_{n}]^{S_{n-1}}\cong\mathbb{C}[H_{n,n-1}]$. Moreover, the composition $\rho_{n}\circ\alpha_{n}:H_{n,n-1}\rightarrow V_{n}/S_{n}$ factors as $H_{n,n-1}\xrightarrow{\rho_{n,n-1}}V_{n}/S_{n-1}\rightarrow V_{n}/S_{n}$.

Proof.

The first claim holds because $V_{n}/S_{n-1}$ is normal and the morphism $\rho_{n,n-1}$ is birational and projective. The second claim is straightforward. ∎

Example 2.6**.**

Let us consider the case of $n=2$. Here $H_{1,\diamondsuit}=V_{2}$ and $H_{2}=\operatorname{Bl}_{\Delta}(V_{2}/S_{2})$. In fact, $H_{n,n-1}=\operatorname{Bl}_{\Delta}(V_{2})$. The morphism $\beta_{2}$ is the natural projection $\operatorname{Bl}_{\Delta}(V_{2})\rightarrow V_{2}$. The morphism $\alpha_{2}$ is the quotient morphism for the action of $S_{2}$ on $\operatorname{Bl}_{\Delta}(V_{2})$ that is induced from $S_{2}$ acting on $V_{2}$.

The morphism $\alpha_{n}\times\beta_{n}:H_{n,n-1}\hookrightarrow H_{n}\times H_{n-1,\diamondsuit}$ is an embedding.

Lemma 2.7**.**

The subvariety $H_{n,n-1}\subset H_{n}\times H_{n-1,\diamondsuit}$ is lagrangian, where we consider $H_{n}$ with the opposite symplectic form.

Proof.

Note that $V^{0}_{n}/S_{n-1}$ is lagrangian in $V^{0}_{n}/S_{n}\times V^{0}_{n}/S_{n-1}$. Recall that $\rho_{n,n-1}$ is an isomorphism over $V^{0}_{n}/S_{n-1}$. Since $H_{n,n-1}$ is irreducible, we see that it is lagrangian in $H_{n}\times H_{n-1,\diamondsuit}$. ∎

Thanks to Example 2.6, we have the following analog of Lemma 2.3.

Lemma 2.8**.**

Let $b\in V_{n}/S_{n}$ be as in Lemma 2.3. Then $\alpha_{n}^{-1}(\rho_{n}^{-1}\left((V_{n}/S_{n})^{\wedge_{b}}\right))$ is the disjoint union of the following schemes:

• (i)

$n-2$* copies of the preimage of $(V_{2}/S_{2}\times V_{n-2})^{\wedge_{0}}$ in $\operatorname{Bl}_{\Delta}(V_{2}/S_{2})\times V_{n-2}$, each mapping to $\rho^{-1}\left((V_{n}/S_{n})^{\wedge_{p}}\right)$ isomorphically,*

• (ii)

and one copy of the preimage of $V_{n}^{\wedge_{0}}$ in $\operatorname{Bl}_{\Delta}(V_{2})\times V_{n-2}$, which maps to $\rho^{-1}\left((V_{n}/S_{n})^{\wedge_{p}}\right)$ via the quotient map for the $S_{2}$-action.

Now we compute $\operatorname{Pic}(H_{n,n-1})$ and $H^{2}(H_{n,n-1},\mathbb{C})$.

Lemma 2.9**.**

Let $n>2$. Then the group $\operatorname{Pic}(H_{n,n-1})$ is a free abelian group with basis $\alpha_{n}^{*}\mathcal{O}_{n}(1)$ and $\beta_{n}^{*}\mathcal{O}_{n-1,\diamondsuit}(1)$. The space $H^{2}(H_{n,n-1},\mathbb{C})$ is two-dimensional with basis

[TABLE]

Proof.

Consider the divisors $D_{1},D_{2}$ in $H_{n,n-1}$ defined as follows: $D_{1}$ is the locus of $(J^{\prime}\subset J)$ such that the support of $\mathbb{C}[x,y]/J^{\prime}$ has repeated points, while $D_{2}$ is the locus, where the support of $J^{\prime}/J$ is contained in that of $\mathbb{C}[x,y]/J^{\prime}$. Note that $D_{1}\cap D_{2}$ has codimension $2$, while $D_{1}\cup D_{2}=H_{n,n-1}\setminus\rho_{n,n-1}^{-1}(V^{0}_{n}/S_{n-1})$, the locus of $(J^{\prime}\subset J)$ such that the support of $J$ has repeated points. Both $D_{1},D_{2}$ are irreducible.

Note that $\beta_{n}$ is smooth at a generic point of $D_{1}$, while from Lemma 2.8 it follows that $\alpha_{n}$ is smooth at the generic points of $D_{1},D_{2}$. Therefore $\beta_{n}^{*}\mathcal{O}_{H_{n-1,\diamondsuit}}(1)=\mathcal{O}(D_{1})$ and $\alpha_{n}^{*}\mathcal{O}_{H_{n}}(1)=\mathcal{O}(D_{1}+D_{2})$. So what remains to prove is that $\operatorname{Cl}(H_{n,n-1})$ is generated by $D_{1},D_{2}$. Then it is freely generated.

Consider the natural action of $T$ on $H_{n,n-1}$. It contains a contracting torus and has finitely many fixed points (because the actions of $T$ on $H_{n-1}$ and $H_{n}$ have these properties and we have $H_{n,n-1}\hookrightarrow H_{n}\times H_{n-1}$). The points in $H_{n,n-1}^{T}$ are labelled by Young diagrams with $n$ boxes and fixed corner box (so that we get a diagram with $n-1$ boxes by removing the fixed box). The $T$-action on the tangent spaces was computed in [C, Section 2]. In particular, take the one-parameter subgroup of $T$ of the form $t\mapsto(t,t^{N})$ for $N\gg 0$. From [C, Proposition 2.6.4] it follows that the open cell for the Bialynicki-Birula decomposition corresponds to the unique fixed point with diagram $(n)$ and there are two codimension $1$ cells corresponding to the two possible corner boxes of $(n-1,1)$. Similarly to the case of usual Hilbert schemes, the closures of these cells are $D_{1}$ and $D_{2}$. So these divisors freely generate $\operatorname{Cl}(H_{n,n-1})$ and the proof for the Picard group is finished. The proof for the second cohomology space follows. ∎

Following Haiman, we will write

[TABLE]

The following result is due to Haiman, [H1, Proposition 3.6.4].

Lemma 2.10**.**

For $n>2$, the canonical bundle $K_{H_{n,n-1}}$ is $\mathcal{O}_{n,n-1}(1,-1)$.

To finish the section, let us note that Proposition 2.4, Example 2.6, Lemmas 2.5, 2.7, 2.8, 2.10, as well as the computation of the Picard group in Lemma 2.9 still hold if we replace $\mathbb{C}$ with a sufficiently large finite localization $\mathsf{R}$ of $\mathbb{Z}$. Hence they also hold over $\mathbb{F}:=\overline{\mathbb{F}}_{p}$ for $p\gg 0$.

2.3. Universal family $U_{n}$

The variety $H_{n}$ parameterizes the codimension $n$ ideals in $\mathbb{C}[x,y]$. In particular, it comes with the sheaf of algebras whose fiber at $I\in H_{n}$ is $\mathbb{C}[x,y]/I$. We denote this sheaf of algebras by $\mathcal{T}_{n}$, this is a rank $n$ vector bundle.

Let $U_{n}$ denote the relative spectrum $\operatorname{Spec}_{\mathcal{O}_{n}}(\mathcal{T}_{n})$. By the construction, $U_{n}$ is a closed subscheme of $H_{n}\times\mathbb{C}^{2}$. It admits a finite degree $n$ morphism to $H_{n}$, to be denoted by $\tau_{n}$ so that $\mathcal{T}_{n}:=\tau_{n*}\mathcal{O}_{U_{n}}$. We have $\mathcal{T}_{n}:=\pi_{*}(\mathcal{O}_{\mu^{-1}(0)^{\theta-s}}\otimes\mathbb{C}^{n})^{G}$, so, in particular, $\mathcal{O}_{n}(1)=\Lambda^{n}\mathcal{T}_{n}$.

Example 2.11**.**

For $n=2$, we have $U_{2}=\operatorname{Bl}_{\Delta}(V_{2})$ and $\tau_{2}$ is the quotient morphism for the $S_{2}$-action. In particular, $U_{2}=H_{2,1}$.

We see that, in general, $U_{n}$ is a Cohen-Macaulay scheme. Note also that $\tau_{n}^{-1}(V_{n}^{0}/S_{n})=V_{n}^{0}/S_{n-1}$. In particular, $U_{n}$ is generically reduced and since it is Cohen-Macaulay, we see that $U_{n}$ is reduced.

Let us write $V_{n}^{1}$ for the locus of points in $V_{n}$ with no more than one pair of repeated points.

Lemma 2.12**.**

The following claims are true:

1. (1)

We have $\alpha_{n}=\tau_{n}\circ\bar{\alpha}_{n}$ for a projective birational morphism $\bar{\alpha}_{n}:H_{n,n-1}\rightarrow U_{n}$. 2. (2)

$\bar{\alpha}_{n}$* is an isomorphism over $V_{n}^{1}/S_{n}$.* 3. (3)

$U_{n}$* is normal and $\alpha_{n}=\tau_{n}\circ\bar{\alpha}_{n}$ is the Stein factorization of $\alpha_{n}$.*

Proof.

To prove (1) we note that $\alpha_{n}$ naturally factors as $H_{n,n-1}\rightarrow H_{n}\times\mathbb{C}^{2}\rightarrow H_{n}$, where the first morphism is projective. That morphism factors as $H_{n,n-1}\rightarrow U_{n}\hookrightarrow H_{n}\times\mathbb{C}^{2}$. We take this arrow for $\bar{\alpha}_{n}$. Over $V^{0}_{n}/S_{n}$, the morphism $\bar{\alpha}_{n}$ is the isomorphism $V^{0}_{n}/S_{n-1}\xrightarrow{\sim}V^{0}_{n}/S_{n-1}$. This proves (1).

Let us prove (2). By Example 2.11, $\bar{\alpha}_{2}$ is an isomorphism. In the notation of Lemma 2.3, $\tau_{n}^{-1}(\rho_{n}^{-1}(V_{n}/S_{n})^{\wedge_{b}})$ admits the same description as $\alpha_{n}^{-1}(\rho_{n}^{-1}(V_{n}/S_{n})^{\wedge_{b}})$, whose description was given Lemma 2.8. This proves (2).

To prove (3), we notice that, by (2), $U_{n}$ is smooth outside of codimension $2$. Since $U_{n}$ is Cohen-Macaulay, it is normal. Hence $\alpha_{n}=\tau_{n}\circ\bar{\alpha}_{n}$ is the Stein factorization. ∎

Lemma 2.13**.**

We have $\mathbb{C}[U_{n}]=\mathbb{C}[V_{n}]^{S_{n-1}}$, which gives rise to the morphism $\bar{\rho}_{n}:U_{n}\rightarrow V_{n}/S_{n-1}$. The morphism $\rho_{n}\circ\tau_{n}:U_{n}\rightarrow V_{n}/S_{n}$ factors as $U_{n}\xrightarrow{\bar{\rho}_{n}}V_{n}/S_{n-1}\rightarrow V_{n}/S_{n}$.

Proof.

By (3) of Lemma 2.12, $\mathbb{C}[U_{n}]=\mathbb{C}[H_{n,n-1}]$. Now the claims of this lemma follow from Lemma 2.5. ∎

Now we are going to describe the restriction of $\mathcal{T}_{n}$ to $\rho_{n}^{-1}((V_{n}/S_{n})^{\wedge_{b}})$, where we use the notation of Lemma 2.3.

Lemma 2.14**.**

We have

[TABLE]

Proof.

Lemma 2.8 together with (2) of Lemma 2.12 gives a description of

[TABLE]

In particular, each of the $n-1$ components of $\tau_{n}^{-1}(\rho_{n}^{-1}((V_{n}/S_{n})^{\wedge_{b}}))$ contributes one copy of $\mathcal{O}_{\rho_{n}^{-1}((V_{n}/S_{n})^{\wedge_{b}})}$ to the restriction of $\mathcal{T}_{n}$. There is one more summand: the sign invariant part of the push-forward from the component (ii) in Lemma 2.8. This is a line bundle that is forced to be $\mathcal{O}_{\rho_{n}^{-1}((V_{n}/S_{n})^{\wedge_{b}})}(1)$ because of the isomorphism $\Lambda^{n}\mathcal{T}_{n}\cong\mathcal{O}_{n}(1)$. ∎

Similarly to the previous section, the results of this one hold over $\mathsf{R}$ and hence over $\mathbb{F}$.

3. Preliminaries on quantizations and Procesi bundles

3.1. Quantizations

Let us start with a general situation. Let $(X,\omega)$ be a smooth symplectic variety over $\mathbb{C}$. In particular, the structure sheaf $\mathcal{O}_{X}$ carries a Poisson bracket $\{\cdot,\cdot\}$ induced by $\omega$. A formal quantization $\mathcal{D}_{\hbar}$ of $X$ is a sheaf of $\mathbb{C}[[\hbar]]$-algebras on $X$ (in the Zariski topology) together with an isomorphism $\kappa:\mathcal{D}_{\hbar}/(\hbar)\xrightarrow{\sim}\mathcal{O}_{X}$ such that

• (a)

$\mathcal{D}_{\hbar}$ is flat over $\mathbb{C}[[\hbar]]$ (i.e., there are no nonzero local sections annihilated by $\hbar$) and complete and separated in the $\hbar$-adic topology (meaning that $\mathcal{D}_{\hbar}\xrightarrow{\sim}\varprojlim_{n\rightarrow+\infty}\mathcal{D}_{\hbar}/(\hbar^{n})$, where the inverse limit is taken in the category of sheaves).

• (b)

$\kappa(\frac{1}{\hbar}[a,b])=\{\kappa(a),\kappa(b)\}$ for any local sections $a,b$ of $\mathcal{D}_{\hbar}$.

Assume now that $X$ carries an action of $\mathbb{C}^{\times}$ such that $t.\omega=t^{d}\omega$. By a grading on $\mathcal{D}_{\hbar}$ we mean an action of $\mathbb{C}^{\times}$ on $\mathcal{D}_{\hbar}$ by sheaf of algebras automorphisms such that $t.\hbar=t^{d}\hbar$ and $\varpi$ is $\mathbb{C}^{\times}$-equivariant. By a graded formal quantization we mean a formal quantization together with a grading. From here we can define a sheaf of filtered algebras in the conical topology (where “open” means Zariski open $\mathbb{C}^{\times}$-stable). Namely, for such a subset $U$, we set $\mathcal{D}(U):=D_{\hbar}(U)_{fin}/(\hbar-1)$, where we write $\mathcal{D}_{\hbar}(U)_{fin}$ for the subalgebra of $\mathbb{C}^{\times}$-finite elements in $\mathcal{D}_{\hbar}(U)$. It is easy to see that the algebras $\mathcal{D}(U)$ form a sheaf in the conical topology to be denoted by $\mathcal{D}$. This sheaf comes with

• •

a complete and separated filtration (induced by the $\hbar$-adic filtration on $\mathcal{D}_{\hbar}$)

• •

and an isomorphism $\operatorname{gr}\mathcal{D}\xrightarrow{\sim}\mathcal{O}_{X}$ of sheaves of graded Poisson algebras in the conical topology (induced by $\kappa$).

By a microlocal filtered quantization of $\mathcal{O}_{X}$ we mean a sheaf $\mathcal{D}$ of algebras in the conical topology on $X$ with the two additional structures above. Then we can recover a graded formal quantization from $\mathcal{D}$ by taking the $\hbar$-adically completed Rees sheaf.

We will need a classification result for the formal graded (equivalently, microlocal filtered) quantizations under certain cohomology vanishing assumptions that was obtained in [L1, Section 2.3] as a ramification of a classification result for all formal quantizations from [BK1]. Namely, in [BK1, Section 4] Bezrukavnikov and Kaledin introduced the noncommutative period map $\mathsf{Per}$ from the set of isomorphism classes of formal quantizations to $H^{2}(X,\mathbb{C})[[\hbar]]$. Further, it was shown in [L1, Section 2.3] that for graded formal quantizations, $\mathsf{Per}$ takes values in $H^{2}(X,\mathbb{C})$, the subspace of elements of $H^{2}(X,\mathbb{C})[[\hbar]]$ independent of $\hbar$.

Lemma 3.1**.**

Assume that $H^{i}(X,\mathcal{O}_{X})=0$ for $i=1,2$. Then $\mathsf{Per}$ is an isomorphism between the set of isomorphism classes of graded formal quantizations and $H^{2}(X,\mathbb{C})$.

Note that if $\mathcal{D}$ is a filtered microlocal quantization of $X$, then $\mathcal{D}^{opp}$ is naturally a filtered microlocal quantization of $X^{opp}$, the same variety as $X$ but with symplectic form $-\omega$. By [L1, Section 2.3], we have

[TABLE]

A microlocal quantization $\mathcal{D}(=\mathcal{D}_{n})$ of $H_{n}$ that we need is constructed in the following way. We first consider the microlocalization $D_{R}$ of the algebra $D(R)$ of differential operators to $T^{*}R$. The sheaf $D_{R}$ is a microlocal filtered quantization of $T^{*}R$. Consider the quantum comoment map $\Phi:\mathfrak{g}\rightarrow D(R)$ given by $\xi\mapsto\xi_{R}$. Then we form the quotient $D_{R}/D_{R}\Phi(\mathfrak{g})$, it is scheme-theoretically supported on $\mu^{-1}(0)$. Recall that $\varphi$ denotes the quotient map $\mu^{-1}(0)^{\theta-s}\rightarrow H_{n}$. We set

[TABLE]

This is a filtered microlocal quantization on $H_{n}$.

Let us now discuss the global sections of $\mathcal{D}$. Let us write $\mathcal{A}_{n}$ for $\mathsf{W}(V_{n})^{S_{n}}$.

Lemma 3.2**.**

We have $R\Gamma(\mathcal{D})=\mathcal{A}_{n}$, i.e., $H^{i}(H_{n},\mathcal{D})=0$ for $i>0$ and $\Gamma(\mathcal{D})=\mathcal{A}_{n}$.

Proof.

The cohomology vanishing part follows from $H^{i}(H_{n},\mathcal{O}_{n})=0$, see Lemma 2.2. By [L1, Lemma 4.2.4], $\Gamma(\mathcal{D})$ coincides with the global Hamiltonian reduction $[D(R)/D(R)\Phi(\mathfrak{g})]^{G}$. The latter is isomorphic to $\mathcal{A}_{n}$ by results of [GG, Section 6]. ∎

Lemma 3.1 applies to $X=H_{n}$. Recall, Section 2.1, that $H^{2}(H_{n},\mathbb{C})$ is identified with $\mathbb{C}$. Consider the symmetrized quantum comoment map $\Phi^{sym}:\mathfrak{g}\rightarrow D(R)$ given by $\Phi^{sym}(\xi)=\frac{1}{2}(\xi_{R}+\xi_{R^{*}})$. Note that $\xi_{R^{*}}=\xi_{R}+\operatorname{tr}(\xi)$ so $\Phi(x)=\Phi^{sym}(\xi)-\frac{1}{2}\operatorname{tr}(\xi)$. By [L1, Section 5], we have

[TABLE]

Now we proceed to discussing quantizations in positive characteristic.

We can do the same construction of quantum Hamiltonian reduction in characteristic $p$ getting a microlocal filtered sheaf of algebras $\mathcal{D}^{\mathbb{F}}$ on $H^{\mathbb{F}}_{n}$. In fact, this can be done over a sufficiently large finite localization $\mathsf{R}$ of $\mathbb{Z}$: we get a microlocal sheaf of algebras $\mathcal{D}^{\mathsf{R}}$ on $H_{n}^{\mathsf{R}}$ with $\mathcal{D}=\mathbb{C}\widehat{\otimes}_{\mathsf{R}}\mathcal{D}^{\mathsf{R}}$ and $\mathcal{D}^{\mathbb{F}}=\mathbb{F}\widehat{\otimes}_{\mathsf{R}}\mathcal{D}^{\mathsf{R}}$ (we take the completed tensor product with respect to the topology defined by the filtration). As was checked in [L4, Lemma 4.4], $R\Gamma(\mathcal{D}^{\mathsf{R}})=\mathcal{A}_{n}^{\mathsf{R}}$. Hence $R\Gamma(\mathcal{D}^{\mathbb{F}})=\mathcal{A}_{n}^{\mathbb{F}}$.

We can also construct a version of a quantization that is a genuine coherent sheaf on the Frobenius twist $H_{n}^{\mathbb{F}(1)}$ following [BFG, Section 4]. Namely, $D(R^{\mathbb{F}})$ is an Azumaya algebra on $T^{*}R^{\mathbb{F}(1)}$. Consider the restriction $D(R^{\mathbb{F}})|_{(\mu^{(1)})^{-1}(0)^{\theta-s}}$. The Lie algebra $\mathfrak{g}^{\mathbb{F}}$ acts on this Azumaya algebra by $\mathcal{O}_{(\mu^{(1)})^{-1}(0)^{\theta-s}}$-linear derivations. It was shown in [BFG, Section 4] that the sheaf of invariants $[D(R^{\mathbb{F}})|_{(\mu^{(1)})^{-1}(0)^{\theta-s}}]^{\mathfrak{g}}$ is also an Azumaya algebra, it is $G^{\mathbb{F}(1)}$-equivariant. Now we can define the sheaf $\underline{\mathcal{D}}^{\mathbb{F}}$ by

[TABLE]

This is a sheaf of Azumaya algebras on $H_{n}^{\mathbb{F}(1)}$ of rank $p^{2n}$. Its restriction to the conical topology comes with a filtration such that $\operatorname{gr}\underline{\mathcal{D}}^{\mathbb{F}}=\operatorname{Fr}_{*}\mathcal{O}^{\mathbb{F}}_{n}$, where we write $\operatorname{Fr}$ for the Frobenius morphism $H_{n}^{\mathbb{F}}\rightarrow H_{n}^{\mathbb{F}(1)}$. So it is a Frobenius-constant quantization in the terminology of [BK2, Section 3.3]. We have $R\Gamma(\underline{\mathcal{D}}^{\mathbb{F}})=\mathcal{A}_{n}^{\mathbb{F}}$ by [BFG, Theorem 4.1.4].

Let us explain how to pass from $\underline{\mathcal{D}}^{\mathbb{F}}$ to $\mathcal{D}^{\mathbb{F}}$. By the construction, we have a homomorphism $\underline{\mathcal{D}}^{\mathbb{F}}\rightarrow\operatorname{Fr}_{*}\mathcal{D}^{\mathbb{F}}$ of sheaves of filtered algebras on $H_{n}^{\mathbb{F}(1)}$. The associated graded homomorphism is the identity. In particular, it is strictly compatible with filtrations. So we see that $\operatorname{Fr}_{*}\mathcal{D}^{\mathbb{F}}$ is the completion of $\underline{\mathcal{D}}^{\mathbb{F}}$.

3.2. Bezrukavnikov-Kaledin construction of Procesi bundles

In [BK2, Section 6], Bezrukavnikov and Kaledin used the Azumaya algebra $\underline{\mathcal{D}}^{\mathbb{F}}$ on $H_{n}^{\mathbb{F}(1)}$ to construct a Procesi bundle. The Azumaya algebra $\underline{\mathcal{D}}^{\mathbb{F}}$ does not split. However, its restriction $\underline{\mathcal{D}}^{\mathbb{F}\wedge}$ to $H_{n}^{\mathbb{F}(1),\wedge}:=\rho_{n}^{-1}((V_{n}^{\mathbb{F}(1)}/S_{n})^{\wedge})$, where $(V_{n}^{\mathbb{F}(1)}/S_{n})^{\wedge}$ denotes the formal neighborhood of [math] in $V_{n}^{\mathbb{F}(1)}/S_{n}$, does. This was proved in [BK2, Section 6.3] (see also [BL, Section 6.2] for a different proof). Let $\hat{\mathcal{E}}^{\mathbb{F}}$ denote a splitting bundle. Note that the endomorphism algebra of $\hat{\mathcal{E}}^{\mathbb{F}}$ is identified with $\mathcal{A}_{n}^{\mathbb{F}\wedge}$ (the completion at $0\in V_{n}^{\mathbb{F}(1)}/S_{n}$). And there are no higher self-extensions of $\hat{\mathcal{E}}^{\mathbb{F}}$, thanks to the formal function theorem and $R^{i}\Gamma(\underline{\mathcal{D}}^{\mathbb{F}})=0$ for $i>0$.

The Azumaya algebra $\mathsf{W}(V_{n}^{\mathbb{F}})^{\wedge}$ admits a standard $S_{n}$-equivariant splitting module leading to a Morita equivalence $\mathcal{A}_{n}^{\mathbb{F}\wedge}\cong\widetilde{A}_{n}^{\mathbb{F}\wedge}$. Here and below we write $\widetilde{A}_{n}^{\mathbb{F}}$ for the algebra $\mathbb{F}[V_{n}]\#S_{n}$. Also $\mathsf{W}(V_{n}^{\mathbb{F}})\#S_{n}$ and $\mathcal{A}_{n}^{\mathbb{F}}$ are Morita equivalent via the bimodule $\mathsf{W}(V_{n}^{\mathbb{F}})$. It follows that the algebras $\mathcal{A}_{n}^{\mathbb{F}\wedge}$ and $\widetilde{A}_{n}^{\mathbb{F}\wedge}$ are Morita equivalent. Let us write $\hat{\mathcal{B}}_{n}^{\mathbb{F}}$ for the Morita equivalence $\widetilde{A}_{n}^{\mathbb{F}\wedge}$-$\mathcal{A}_{n}^{\mathbb{F}\wedge}$-bimodule, and $\hat{\mathcal{B}}_{n}^{o,\mathbb{F}}$ for the inverse bimodule. In particular, we have

[TABLE]

We set (basically following [L5, Section 4.3])

[TABLE]

Since $\hat{\mathcal{B}}_{n}^{\mathbb{F}}$ is projective as a right $\mathcal{A}_{n}^{\mathbb{F}\wedge}$-module, we see that $\hat{\mathcal{P}}^{\mathbb{F}}_{n}$ has no higher self-extensions and

[TABLE]

Also, since $\hat{\mathcal{P}}^{\mathbb{F}}_{n}$ has no higher self-extensions, it admits a $T_{c}^{\mathbb{F}(1)}$-equivariant structure, see [V]. Since the $T_{c}^{\mathbb{F}(1)}$-action on $H_{n}^{\mathbb{F}(1)}$ is contracting, the completion functor is an equivalence

[TABLE]

Let $\mathcal{P}^{\mathbb{F}}_{n}$ be the $T_{c}^{\mathbb{F}(1)}$-equivariant sheaf on $H_{n}^{\mathbb{F}(1)}$ corresponding to $H_{n}^{\mathbb{F}(1)\wedge}$. Note that as a sheaf $\mathcal{P}_{n}^{\mathbb{F}}$ is independent of the choice of an equivariant structure on $\hat{\mathcal{P}}_{n}^{\mathbb{F}}$. Moreover, $\mathcal{P}_{n}^{\mathbb{F}}$ has no higher self-extensions, and we have $\operatorname{End}(\mathcal{P}_{n}^{\mathbb{F}})\cong\widetilde{A}_{n}^{\mathbb{F}(1)}$. We can twist $\mathcal{P}_{n}^{\mathbb{F}}$ with a line bundle and achieve that $\left(\mathcal{P}_{n}^{\mathbb{F}}\right)^{S_{n}}=\mathcal{O}_{n}^{\mathbb{F}(1)}$.

Let us also point out that, since $H_{n}^{\mathbb{F}}$ is defined over $\mathbb{F}_{p}$, we have $H_{n}^{\mathbb{F}(1)}\cong H_{n}^{\mathbb{F}}$. So we can view $\mathcal{P}_{n}^{\mathbb{F}}$ as a vector bundle over $H_{n}^{\mathbb{F}}$.

Then one can lift $\mathcal{P}_{n}^{\mathbb{F}}$ to characteristic [math] as described in [BK2, Sections 2.3, 6.4] getting a Procesi bundle $\mathcal{P}_{n}$ on $H_{n}$.

3.3. Procesi bundles, axiomatically

By a Procesi bundle on $H_{n}$ we mean a $T_{c}$-equivariant vector bundle $\mathcal{P}$ together with a graded $\mathbb{C}[V_{n}]^{S_{n}}$-algebra isomorphism $\operatorname{End}(\mathcal{P})\xrightarrow{\sim}\widetilde{A}_{n}^{opp}$ satisfying

• (i)

$\operatorname{Ext}^{i}(\mathcal{P},\mathcal{P})=0$ for all $i>0$,

By an isomorphism of two Procesi bundles $\mathcal{P},\mathcal{P}^{\prime}$ we mean a $\mathbb{C}^{\times}$-equivariant isomorphism $\mathcal{P}\xrightarrow{\sim}\mathcal{P}^{\prime}$ such that the corresponding isomorphism $\widetilde{A}_{n}\xrightarrow{\sim}\widetilde{A}_{n}$ is inner (this algebra has a graded outer automorphism given by $\alpha\mapsto\alpha,\sigma\mapsto(-1)^{\sigma}\sigma$ for $\alpha\in V^{*}_{n},\sigma\in S_{n}$). Thanks to the isomorphism $\operatorname{End}(\mathcal{P})\xrightarrow{\sim}\widetilde{A}_{n}^{opp}$, every fiber of $\mathcal{P}$ is the regular representation of $S_{n}$. It follows that $\mathcal{P}^{S_{n}}$ (as well as the sign component $\mathcal{P}^{\operatorname{sgn}}$) is a line bundle.

By a normalized Procesi bundle we mean a Procesi bundle satisfying the additional condition:

• (ii)

$\mathcal{P}^{S_{n}}\cong\mathcal{O}_{n}$ (an isomorphism of $T_{c}$-equivariant vector bundles).

We can twist every Procesi bundle with a line bundle (and take the induced isomorphism $\operatorname{End}(\mathcal{P})\xrightarrow{\sim}\widetilde{A}_{n}$) to achieve (ii).

In particular, the Bezrukavnikov-Kaledin bundle $\mathcal{P}_{n}$ recalled in Section 3.2, is a normalized Procesi bundle.

The following proposition is a special case of [L2, Theorems 1.1,1.2]. Note that we have an anti-involution of $\tilde{A}_{n}$ defined by $\alpha\mapsto\alpha,\sigma\mapsto\sigma^{-1}$. So the dual of a (normalized) Procesi bundle is also a (normalized) Procesi bundle.

Proposition 3.3**.**

The following statements hold.

1. (1)

There are exactly two distinct normalized Procesi bundle on $H_{n}$. 2. (2)

They are dual to each other. 3. (3)

For one of these Procesi bundles we have $\mathcal{P}^{S_{n-1}}\cong\mathcal{T}_{n}$.

The following lemma can be proved similarly to (1),(3) of Proposition 3.3, but we will deduce it from this proposition and another result of [L2].

Lemma 3.4**.**

If $\mathcal{P}^{S_{n-1}}\cong\mathcal{T}_{n}$, then $\mathcal{P}^{\operatorname{sgn}}\cong\mathcal{O}_{n}(1)$.

Proof.

Let $b\in V_{n}/S_{n}$ be as in Lemma 2.3. By [L2, Proposition 4.1], the restriction of $\mathcal{P}$ to $\rho_{n}^{-1}((V_{n}/S_{n})^{\wedge_{b}})$ is isomorphic to the direct sum of $n!/2$ copies of a Procesi bundle on $\rho_{2}^{-1}((V_{2}/S_{2})^{\wedge_{0}})$. The latter is isomorphic to $\mathcal{O}^{\wedge}_{2}\oplus\mathcal{O}^{\wedge}_{2}(1)$ or $\mathcal{O}^{\wedge}_{2}\oplus\mathcal{O}^{\wedge}_{2}(-1)$. Since $\mathcal{P}^{S_{n-1}}\cong\mathcal{T}_{n}$, Lemma 2.14 implies that the restriction is $(\mathcal{O}_{2}^{\wedge}\oplus\mathcal{O}_{2}^{\wedge}(1))^{\oplus n!/2}$. But the restriction of $\mathcal{O}_{n}(k)$ is $\mathcal{O}_{2}^{\wedge}(k)$ for all $k\in\mathbb{Z}$. It follows that $\mathcal{P}^{\operatorname{sgn}}\cong\mathcal{O}$ or $\mathcal{O}(1)$. Let us show that the former is impossible. Indeed, under the derived equivalence $R\operatorname{Hom}_{\mathcal{O}_{H_{n}}}(\mathcal{P},\bullet)$, the bundles $\mathcal{P}^{S_{n}},\mathcal{P}^{sgn}$ both map to $\mathbb{C}[V]$ but in the first case the action of $S_{n}$ is the natural one, and in the second case it is twisted with the sign. It is easy to see that these two $\tilde{A}_{n}$-modules are not isomorphic, which implies $\mathcal{P}^{\operatorname{sgn}}\not\cong\mathcal{P}^{S_{n}}$. So $\mathcal{P}^{\operatorname{sgn}}\cong\mathcal{O}_{n}(1)$. ∎

Corollary 3.5**.**

Let $\mathcal{P}$ be a Procesi bundle satisfying $\mathcal{P}^{S_{n-1}}\cong\mathcal{T}_{n}$. Then $\mathcal{P}\cong\mathcal{P}^{*}(1)$, a $\mathbb{C}[V_{n}]$-linear isomorphism of vector bundles on $H_{n}$.

Proof.

Twist the action of $\widetilde{A}_{n}$ on $\mathcal{P}^{*}$ with the outer automorphism mentioned above. Then $\mathcal{P}^{*}(1)$ becomes normalized and also the sign components in $\mathcal{P},\mathcal{P}^{*}(1)$ are isomorphic. A $\mathbb{C}[V_{n}]$-linear isomorphism $\mathcal{P}\cong\mathcal{P}^{*}(1)$ now follows from Proposition 3.3 and Lemma 3.4. ∎

Remark 3.6**.**

We can define the notion of an abstract Procesi bundle over $\mathbb{F}(=\overline{\mathbb{F}}_{p})$ for $p\gg 0$. The arguments of the proofs of [L2, Theorem 1.1,1.2] carry over to this case without any significant modifications. Since $\mathcal{P}^{\operatorname{sgn}}$ is obtained by lifting its characteristic $p$ counterpart to characteristic [math], Lemma 3.4 continues to hold over $\mathbb{F}$. So does Corollary 3.5.

3.4. Quantization of lagrangian subvarieties

In this section we will explain results from [BGKP] on quantizations of the structure sheaf of a smooth lagrangian subvariety $Y$ in a smooth symplectic variety $X$ over $\mathbb{C}$. Let $\mathcal{D}_{\hbar}$ be a formal quantization of $\mathcal{O}_{X}$. We want to know when there is a coherent sheaf $\mathcal{M}_{\hbar}$ of $\mathcal{D}_{\hbar}$-modules that is flat over $\mathbb{C}[[\hbar]]$ and comes with an identification $\mathcal{M}_{\hbar}/\hbar\mathcal{M}_{\hbar}\cong\mathcal{O}_{Y}$ of $\mathcal{O}_{X}$-modules. Then we call $\mathcal{M}_{\hbar}$ a formal quantization of $Y$.

Let $\iota$ denote the inclusion $Y\hookrightarrow X$ so that we have the pull-back map $\iota^{*}:H^{2}(X,\mathbb{C})\rightarrow H^{2}(Y,\mathbb{C})$. Now assume that $H^{i}(Y,\mathcal{O}_{Y})=0$ for $i=1,2$.

Recall that we have the period $\mathsf{Per}(\mathcal{D}_{\hbar})\in H^{2}(X,\mathbb{C})[[\hbar]]$. The following is a special case of [BGKP, Theorem 1.1.4].

Proposition 3.7**.**

The $\mathcal{O}_{X}$-module $\mathcal{O}_{Y}$ admits a formal quantization $\mathcal{M}_{\hbar}$ if and only if

[TABLE]

Note that this is true over any characteristic field [math] field, which is already the generality of [BGKP].

We will be interested in microlocal filtered quantizations of $\mathcal{O}_{Y}$. Let $\mathcal{D}$ be a microlocal filtered quantization of $X$. Let $\mathcal{M}$ be a coherent sheaf of $\mathcal{D}$-modules. By a good filtration on $\mathcal{M}$ we mean a $\mathcal{D}$-module filtration $\mathcal{M}=\bigcup_{i\in\mathbb{Z}}\mathcal{M}_{\leqslant i}$ by sheaves of vector spaces subject to the following two conditions:

• •

The filtration is complete and separated.

• •

The associated graded $\mathcal{O}_{X}$-module $\operatorname{gr}\mathcal{M}$ is coherent.

For example, if $\mathcal{M}_{\hbar}$ is a coherent $\mathcal{D}_{\hbar}$-module that comes with a $\mathbb{C}^{\times}$-action compatible with that on $\mathcal{D}_{\hbar}$, then we can produce a coherent $\mathcal{D}$-module $\mathcal{M}:=\mathcal{M}_{\hbar,fin}/(\hbar-1)\mathcal{M}_{\hbar,fin}$. This module comes with a natural filtration, which is good.

A good filtration is far from being unique. However, any two good filtrations $(\mathcal{M}_{\leqslant i})_{i\in\mathbb{Z}}$ and $(\mathcal{M}_{\preceq i})_{i\in\mathbb{Z}}$ of $\mathcal{M}$ are compatible in the following sense.

Lemma 3.8**.**

There are integers $d_{1},d_{2}$ such that $\mathcal{M}_{\leqslant i-d_{1}}\subset\mathcal{M}_{\preceq i}\subset\mathcal{M}_{\leqslant i+d_{2}}$ for all $i$.

Proof.

This can be checked locally, where we deal with good filtrations on modules over algebras. In this case, the claim is classical. ∎

We have the following corollary of Proposition 3.7.

Corollary 3.9**.**

We assume that $X$ comes with a $T\times\mathbb{C}^{\times}$-action, where a torus $T$ preserves $\omega$, and $\mathbb{C}^{\times}$ rescales $\omega$ as before. Let $\mathcal{D}$ be a microlocal filtered quantization of $X$ with an action of a torus $T$ by filtered algebra automorphisms. Assume that $Y$ is $T$-stable and there is a formal quantization of $Y$. Then there is a filtered quantization of $\mathcal{O}_{Y}$ to a coherent $\mathcal{D}$-module $\mathcal{M}$ that carries an action of $T$ that lifts the $T$-action on $\mathcal{O}_{Y}$ and is compatible with the action of $T$ on $\mathcal{D}$.

Proof.

Similarly to [L1, Section 2.3], one shows that the existence of a $T\times\mathbb{C}^{\times}$-action on $\mathcal{M}_{\hbar}$ with required properties is equivalent to the claim that the isomorphism class of the formal quantization $\mathcal{M}_{\hbar}$ is fixed by $T\times\mathbb{C}^{\times}$. Let $\mathsf{Quant}(Y,\mathcal{D}_{\hbar})$ denote the set of isomorphism classes of formal quantizations of $\mathcal{O}_{Y}$.

According to (2) of [BGKP, Theorem 1.1.4], $\mathsf{Quant}(Y,\mathcal{D}_{\hbar})$ is a torsor over the group $\mathfrak{F}$ of $\mathcal{O}_{Y}[[\hbar]]^{\times}$-torsors with a flat connection that are isomorphic to the principal $\mathbb{G}_{m}$-bundle associated to $\mathcal{O}_{Y}$ modulo $\hbar$. Note that $\mathfrak{F}$ is a vector space that is realized as $\varprojlim_{k\rightarrow\infty}\mathfrak{F}_{k}$, where $\mathfrak{F}_{k}$ is the similarly defined space of flat $(\mathcal{O}_{Y}[[\hbar]]/(\hbar^{k}))^{\times}$-torsors. Each of the spaces $\mathfrak{F}_{k}$ comes with a rational action of $T\times\mathbb{C}^{\times}$. Moreover, for any element $\delta\in\mathsf{Quant}(Y,\mathcal{D}_{\hbar})$, the map $t\mapsto\phi_{k}(t):T\times\mathbb{C}^{\times}\rightarrow\mathfrak{F}_{k}$ defined by $\phi_{k}(t)\delta=t.\delta$ is algebraic. It follows that the cocycle $t\mapsto\phi(t)$ is actually a coboundary. Equivalently, there is a $T\times\mathbb{C}^{\times}$-fixed point in $\mathsf{Quant}(Y,\mathcal{D}_{\hbar})$.

∎

4. Quantization of the nested Hilbert scheme

Recall that we write $H_{n-1,\diamondsuit}$ for $H_{n-1}\times\mathbb{A}^{2}$. In Section 3.1 we have constructed the filtered microlocal quantizations of $H_{n},H_{n-1}$ to be denoted by $\mathcal{D}_{n},\mathcal{D}_{n-1}$, both have period $-\frac{1}{2}$. We also write $\mathcal{D}_{n-1,\diamondsuit}$ for $\mathcal{D}_{n-1}\widehat{\otimes}\mathsf{W}(V_{2})$, this is a filtered microlocal quantization of $H_{n-1,\diamondsuit}$.

In this section we study a quantization of the $\mathcal{O}_{n}$-$\mathcal{O}_{n-1,\diamondsuit}$-bimodule $\mathcal{O}_{n,n-1}$ to a filtered coherent $\mathcal{D}_{n-1,\diamondsuit}$-$\mathcal{D}_{n}$-bimodule to be denoted by $\mathcal{D}_{n,n-1}$.

In Section 4.1, we consider the quantizations over $\mathbb{C}$ and over $\mathbb{Q}$ using results recalled in Section 3.4. We will see that the filtered $\Gamma(\mathcal{D}_{n-1,\diamondsuit})$-$\Gamma(\mathcal{D}_{n})$-bimodule of global sections is equal to $\mathcal{A}_{n-1,\diamondsuit}$ and also show how to recover $\mathcal{D}_{n,n-1}$ from its global sections.

In Section 4.2, we will produce a form of $\mathcal{D}_{n,n-1}$ over a finite localization $\mathsf{R}$ of $\mathbb{Z}$ and show that it is still a microlocal filtered quantization of $\mathcal{O}^{\mathsf{R}}_{n,n-1}$.

In Section 4.3, we reduce the $\mathsf{R}$-form from Section 4.2 mod $p$ for $p\gg 0$. We pass from the resulting microlocal quantization $\mathcal{D}^{\mathbb{F}}_{n,n-1}$ to a “Frobenius constant” quantization $\underline{\mathcal{D}}_{n,n-1}^{\mathbb{F}}$ that turns out to be a splitting bundle for the Azumaya algebra $\left(\underline{\mathcal{D}}^{\mathbb{F},opp}_{n}\otimes\underline{\mathcal{D}}^{\mathbb{F}}_{n-1,\diamondsuit}\right)|_{H_{n,n-1}^{\mathbb{F}(1)}}.$

4.1. Quantization in characteristic [math]

Let the base field be $\mathbb{C}$. From (3.1) and $\mathsf{Per}(\mathcal{D}_{n-1})=-\frac{1}{2}$, we get $\mathsf{Per}(\mathcal{D}_{n-1}^{\diamondsuit,opp})=\frac{1}{2}$. By Lemma 2.10, for $n>2$, we have

[TABLE]

where we write $\iota$ for the inclusion $H_{n,n-1}\hookrightarrow H_{n}\times H_{n-1,\diamondsuit}$. It is lagrangian by Lemma 2.7.

The following lemma establishes the remaining assumption from Section 3.4.

Lemma 4.1**.**

We have $H^{i}(H_{n,n-1},\mathcal{O})=0$ for $i>0$ and $\mathbb{C}[H_{n,n-1}]=\mathbb{C}[V_{n}]^{S_{n-1}}$.

Proof.

Recall the morphism $\rho_{n,n-1}:H_{n,n-1}\rightarrow V_{n}/S_{n-1}$, see Lemma 2.5. It is a birational proojective morphism. The variety $V_{n}/S_{n-1}$ has rational singularities. It follows that $\rho_{n,n-1*}\mathcal{O}_{n,n-1}=\mathcal{O}_{V_{n}/S_{n-1}}$. ∎

Now using Corollary 3.9 we see that there is a filtered coherent $\mathcal{D}_{n}^{opp}\widehat{\otimes}\mathcal{D}_{n-1,\diamondsuit}$-module $\mathcal{D}_{n,n-1}$ with a good filtration satisfying $\operatorname{gr}\mathcal{D}_{n,n-1}=\mathcal{O}_{n,n-1}$. We can also assume that the $T_{h}$-action lifts from $\mathcal{O}_{n,n-1}$ to $\mathcal{D}_{n,n-1}$. Thanks to Lemma 4.1, we have a $T_{h}$-equivariant filtered isomorphism $\operatorname{gr}\Gamma(\mathcal{D}_{n,n-1})=\mathbb{C}[V_{n}]^{S_{n-1}}$.

Proposition 4.2**.**

We have an isomorphism $\Gamma(\mathcal{D}_{n,n-1})\cong\mathcal{A}_{n-1,\diamondsuit}$ of filtered $\mathcal{A}_{n-1,\diamondsuit}$-$\mathcal{A}_{n}$- bimodules.

Proof.

Note that $\mathcal{A}_{n}$ embeds into $\mathcal{A}_{n-1,\diamondsuit}$ so we can view $\Gamma(\mathcal{D}_{n,n-1})$ as an $\mathcal{A}_{n}$-bimodule. The filtration 0 component of $\Gamma(\mathcal{D}_{n,n-1})$ is one-dimensional thanks to the isomorphism $\operatorname{gr}\Gamma(\mathcal{D}_{n,n-1})\cong\mathbb{C}[V_{n}]^{S_{n-1}}$. We start by proving that this component is centralized by $\mathcal{A}_{n}$.

Let $a$ denote a nonzero element in the filtration component of degree [math] in $\Gamma(\mathcal{D}_{n,n-1})$. This element is $T_{h}$-invariant. The algebra $\mathcal{A}_{n}$ is generated by its subalgebras $\mathbb{C}[\underline{x}]^{S_{n}},\mathbb{C}[\underline{y}]^{S_{n}}$, see, e.g., [W], so it is enough to show that $a$ commutes with $\mathbb{C}[\underline{x}]^{S_{n}}$ and $\mathbb{C}[\underline{y}]^{S_{n}}$. We will do this for the first subalgebra, the second is analogous. Let $F\in\mathbb{C}[\underline{x}]^{S_{n}}$ be a homogeneous element of degree $m$. Then it lies in the filtration degree $m$ component of $\mathcal{A}_{n}$ and also in the degree $m$ component for the grading induced by the $T_{h}$-action. It follows that $[F,a]$ lies in the filtration component $m-2$ and in the grading component $m$. However, since $\operatorname{gr}\Gamma(\mathcal{D}_{n,n-1})=\mathbb{C}[V_{n}]^{S_{n-1}}$, we see that the intersection of that filtration component and that grading component is zero. So $[F,a]=0$.

Consider the map $\zeta:\mathcal{A}_{n-1,\diamondsuit}\rightarrow\Gamma(\mathcal{D}_{n,n-1}),b\mapsto ab$. The associated graded morphism of $\zeta$ is an isomorphism of graded modules, so $\zeta$ is an isomorphism strictly compatible with the filtrations. Since $a$ commutes with $\mathcal{A}_{n}$, we see that $\iota$ is an isomorphism of bimodules. ∎

Now let us recover $\mathcal{D}_{n,n-1}$ (without the filtration) from the bimodule $\mathcal{A}_{n-1,\diamondsuit}$.

Proposition 4.3**.**

We have the following isomorphism of $\mathcal{D}_{n-1,\diamondsuit}$-$\mathcal{D}_{n}$-bimodules

[TABLE]

Proof.

We claim that the functors $\Gamma$ and

[TABLE]

are mutually inverse equivalences between the category of coherent $\mathcal{D}_{n-1,\diamondsuit}\widehat{\otimes}\mathcal{D}_{n}^{opp}$-modules and the category of finitely generated $\mathcal{A}_{n-1,\diamondsuit}\otimes\mathcal{A}_{n}^{opp}$-modules. This claim is usually called an abelian localization theorem. This and Proposition 4.2 imply (4.3).

Our claim that $\Gamma,\operatorname{Loc}$ are mutually inverse equivalences is pretty classical and can be proved in several different ways. For example, it is a special case of the main result [GS] (which is based upon Haiman’s work so we cannot use that) or of [L3] (which is independent from Haiman’s work). Alternatively, it is a direct corollary of the main results of [MN1] and [MN2]. Namely, the algebra $\mathcal{A}_{n-1,\diamondsuit}\otimes\mathcal{A}_{n}^{opp}$ has finite homological dimension, so $R\Gamma$ and $L\operatorname{Loc}$ are quasi-inverse derived equivalences by [MN1, Theorem 1.1]. By [MN2, Corollary 1.3, Section 8], the functor $\Gamma$ is exact, so is an equivalence of abelian categories. Yet alternatively, this follows from results of [BEG] (that rational Cherednik algebras with integral parameters are simple) and the general results on the abelian localization theorem from [BPW, Section 5.3] and [BL, Section 4.2]. ∎

Now recall, Section 3.4, that $\mathcal{D}_{n,n-1}$ is defined over $\mathbb{Q}$, let us denote the corresponding $\mathbb{Q}$-form by $\mathcal{D}^{\mathbb{Q}}_{n,n-1}$. The proof of Proposition 4.2 shows that $\Gamma(\mathcal{D}_{n,n-1}^{\mathbb{Q}})=\mathcal{A}_{n-1,\diamondsuit}^{\mathbb{Q}}$, an isomorphism of $\mathcal{A}^{\mathbb{Q}}_{n-1,\diamondsuit}$-$\mathcal{A}_{n}^{\mathbb{Q}}$-bimodules. Proposition 4.3 also holds over $\mathbb{Q}$ because (4.1) is defined over $\mathbb{Q}$.

Remark 4.4**.**

As we have seen, the varieties $H_{n},H_{n-1,\diamondsuit}$ as well as the quantizations $\mathcal{D}_{n},\mathcal{D}_{n-1,\diamondsuit}$ can be constructed via (quantum) Hamiltonian reductions. The variety $H_{n,n-1}$ also has a “Hamiltonian reduction” description. It would be interesting to find such a description for $\mathcal{D}_{n,n-1}$.

4.2. $\mathsf{R}$-form

As was mentioned in Section 2.1, the schemes $H_{n},H_{n-1}$ are defined over a finite localization $\mathsf{R}$ of $\mathbb{Z}$. We assume that $n!$ is invertible in $\mathsf{R}$ so taking the $S_{n}$-invariants behaves in the usual way. The same is true for $H_{n,n-1}$. Further localizing finitely many elements in $\mathsf{R}$, we can assume that the $\mathsf{R}$-schemes $H^{\mathsf{R}}_{n},H^{\mathsf{R}}_{n-1},H^{\mathsf{R}}_{n,n-1}$ are regular and $R\Gamma(\mathcal{O}_{i}^{\mathsf{R}})=\mathsf{R}[V_{i}]^{S_{i}}$ for $i=n,n-1$, $R\Gamma(\mathcal{O}^{\mathsf{R}}_{n,n-1})=\mathsf{R}[V_{n}]^{S_{n-1}}$.

As was mentioned in Section 3.1, we can further assume that the microlocal filtered quantizations $\mathcal{D}_{n},\mathcal{D}_{n-1}$ are defined over $\mathsf{R}$ and that the corresponding $\mathsf{R}$-forms $\mathcal{D}_{n}^{\mathsf{R}},\mathcal{D}_{n-1}^{\mathsf{R}}$ are filtered microlocal quantizations of $\mathcal{O}_{n}^{\mathsf{R}},\mathcal{O}_{n-1}^{\mathsf{R}}$. By an $\mathsf{R}$-form (in the case of $\mathcal{D}_{n}$, for example), we mean a subsheaf $\mathcal{D}_{n}^{\mathsf{R}}\subset\mathcal{D}_{n}^{\mathbb{Q}}$ of $\mathsf{R}$-algebras such that the inclusion $\mathcal{D}_{n}^{\mathsf{R}}\hookrightarrow\mathcal{D}_{n}^{\mathbb{Q}}$ gives an isomorphism of filtered sheaves of algebras $\mathbb{Q}\widehat{\otimes}_{\mathsf{R}}\mathcal{D}_{n}^{\mathsf{R}}\xrightarrow{\sim}\mathcal{D}_{n}^{\mathbb{Q}}$.

Our goal in this section is to prove the following technical result.

Lemma 4.5**.**

After replacing $\mathsf{R}$ with a finite localization, there is an $\mathsf{R}$-form $\mathcal{D}_{n,n-1}^{\mathsf{R}}$ of $\mathcal{D}_{n,n-1}^{\mathbb{Q}}$ such that the following conditions hold:

1. (1)

$\mathcal{D}_{n,n-1}^{\mathsf{R}}$* is a filtered $\mathcal{D}_{n-1,\diamondsuit}^{\mathsf{R}}$-$\mathcal{D}_{n}^{\mathsf{R}}$-bimodule with respect to the filtration restricted from $\mathcal{D}_{n,n-1}^{\mathbb{Q}}$,* 2. (2)

the filtration on $\mathcal{D}_{n,n-1}^{\mathsf{R}}$ is good, 3. (3)

and we have an isomorphism $\operatorname{gr}\mathcal{D}_{n,n-1}^{\mathsf{R}}\xrightarrow{\sim}\mathcal{O}_{n,n-1}^{\mathsf{R}}$ of graded sheaves (in the conical topology) of bimodules. 4. (4)

$R\Gamma(\mathcal{D}_{n,n-1}^{\mathsf{R}})=\mathcal{A}_{n-1,\diamondsuit}^{\mathsf{R}}$, the equality of subbimodules of $R\Gamma(\mathcal{D}_{n,n-1}^{\mathbb{Q}})=\mathcal{A}_{n-1,\diamondsuit}^{\mathbb{Q}}$.

Proof.

Consider the $\mathcal{D}_{n-1,\diamondsuit}^{\mathsf{R}}$-$\mathcal{D}_{n}^{\mathsf{R}}$-bimodule

[TABLE]

The bimodule $\mathcal{D}_{n,n-1}^{\prime\mathsf{R}}$ admits a natural homomorphism of sheaves of bimodules, say $\iota$, to $\mathcal{D}_{n,n-1}^{\mathbb{Q}}$ and comes with the tensor product filtration, which is good. It follows from Proposition 4.3 that $\iota$ induces an isomorphism $\mathbb{Q}\widehat{\otimes}_{\mathsf{R}}\mathcal{D}^{\prime\mathsf{R}}_{n,n-1}\xrightarrow{\sim}\mathcal{D}^{\mathbb{Q}}_{n,n-1}$ of sheaves of bimodules. Consider $\ker\iota\subset\mathcal{D}_{n,n-1}^{\prime\mathsf{R}}$. We see that $\operatorname{gr}\ker\iota$ is a coherent sheaf on $H_{n}^{\mathsf{R}}\times H_{n-1,\diamondsuit}^{\mathsf{R}}$ that is $\mathsf{R}$-torsion. So after replacing $\mathsf{R}$ with a finite localization, $\iota$ becomes injective. Hence $\mathcal{D}^{\prime\mathsf{R}}_{n,n-1}$ is an $\mathsf{R}$-form of $\mathcal{D}^{Q}_{n,n-1}$

Note that $\mathcal{D}_{n,n-1}^{\prime\mathsf{R}}$ also satisfies (1). However, it does not need to satisfy (3): even the filtration on $\mathcal{D}_{n,n-1}^{\mathbb{Q}}$ induced from $\mathcal{D}_{n,n-1}^{\prime\mathsf{R}}$ does not need to coincide with the initial filtration on $\mathcal{D}_{n,n-1}^{\mathbb{Q}}$. But both filtrations on $\mathcal{D}_{n,n-1}^{\mathbb{Q}}$ are good. We will show that after replacing $\mathsf{R}$ with a finite localization there is a good filtration on $\mathcal{D}^{\prime\mathsf{R}}_{n,n-1}$ satisfying (3). For this, consider the completed Rees sheaf $\mathcal{D}^{\prime\mathsf{R}}_{n,n-1,\hbar}$. Then the completed Rees sheaf $\mathcal{D}^{\prime\mathbb{Q}}_{n,n-1,\hbar}$ is obtained as $\mathbb{Q}\widehat{\otimes}_{\mathsf{R}}\mathcal{D}^{\prime\mathsf{R}}_{n,n-1,\hbar}$.

Recall, Lemma 3.8, that the good filtration on $\mathcal{D}_{n,n-1}^{\mathbb{Q}}$ is squeezed between appropriate shifts of the good filtration of $\mathcal{D}^{\prime\mathbb{Q}}_{n,n-1}$. So $\mathcal{D}^{\mathbb{Q}}_{n,n-1,\hbar}$ embeds into $\mathcal{D}^{\prime\mathbb{Q}}_{n,n-1,\hbar}$. The image contains $\hbar^{N}\mathcal{D}^{\prime\mathbb{Q}}_{n,n-1,\hbar}$ for a sufficiently large integer $N$ and (after shifting the filtrations) we can assume that the embedding is $T_{c}$-equivariant. Note that $\mathcal{D}^{\prime\mathbb{Q}}_{n,n-1,\hbar}/\hbar^{N}\mathcal{D}^{\prime\mathbb{Q}}_{n,n-1,\hbar}=\mathbb{Q}\otimes_{\mathsf{R}}\left(\mathcal{D}^{\prime\mathsf{R}}_{n,n-1,\hbar}/\hbar^{N}\mathcal{D}^{\prime\mathsf{R}}_{n,n-1,\hbar}\right)$ (where we no longer need to complete the tensor product). After replacing $\mathsf{R}$ with a finite localization, the subbimodule $\mathcal{D}^{\mathbb{Q}}_{n,n-1,\hbar}/\hbar^{N}\mathcal{D}^{\prime\mathbb{Q}}_{n,n-1,\hbar}$ becomes defined over $\mathsf{R}$ and so gives rise to a $T_{c}$-stable subbimodule $\mathcal{D}^{\mathsf{R}}_{n,n-1,\hbar}\subset\mathcal{D}^{\prime\mathsf{R}}_{n,n-1,\hbar}$ containing $\hbar^{N}\mathcal{D}^{\prime\mathsf{R}}_{n,n-1,\hbar}$. From $\mathcal{D}^{R}_{n,n-1,\hbar}$ we produce a filtered coherent $\mathcal{D}^{\mathsf{R}}_{n-1,\diamondsuit}$-$\mathcal{D}^{\mathsf{R}}_{n}$-bimodule $\mathcal{D}^{\mathsf{R}}_{n,n-1}$ with $\mathbb{Q}\widehat{\otimes}_{\mathsf{R}}\mathcal{D}^{\mathsf{R}}_{n,n-1}\xrightarrow{\sim}\mathcal{D}^{\mathbb{Q}}_{n,n-1}$. Hence $\mathbb{Q}\otimes_{\mathsf{R}}\operatorname{gr}\mathcal{D}^{\mathsf{R}}_{n,n-1}\xrightarrow{\sim}\mathcal{O}^{\mathbb{Q}}_{n,n-1}$. After replacing $\mathsf{R}$ with a finite localization again, we achieve that (3) holds. Similarly, we achieve that (4) holds. ∎

4.3. Quantization in characteristic $p$

Set $\mathbb{F}:=\overline{\mathbb{F}}_{p}$, where $\mathbb{F}_{p}$ is a quotient of $\mathsf{R}$ (so that $p$ is very large). Set $\mathcal{D}^{\mathbb{F}}_{n,n-1}:=\mathbb{F}\widehat{\otimes}_{\mathsf{R}}\mathcal{D}^{\mathsf{R}}_{n,n-1}$. This is a microlocal filtered $\mathcal{D}^{\mathbb{F}}_{n-1,\diamondsuit}$-$\mathcal{D}_{n}^{\mathbb{F}}$-bimodule.

Lemma 4.6**.**

The bimodule $\mathcal{D}^{\mathbb{F}}_{n,n-1}$ is a microlocal filtered quantization of $\mathcal{O}_{n,n-1}^{\mathbb{F}}$. Further, we have $R\Gamma(\mathcal{D}^{\mathbb{F}}_{n,n-1})=\mathcal{A}_{n-1,\diamondsuit}^{\mathbb{F}}$, an isomorphism of filtered bimodules.

Proof.

This follows from Lemma 4.5 because $H_{n,n-1}^{\mathsf{R}}$ is flat over $\operatorname{Spec}(\mathsf{R})$, while $\mathcal{A}_{n-1,\diamondsuit}^{\mathsf{R}}$ is flat over $\mathsf{R}$. ∎

Now we are going to produce a coherent sheaf $\underline{\mathcal{D}}^{\mathbb{F}}_{n,n-1}$ on $H_{n}^{\mathbb{F}(1)}\times H_{n-1,\diamondsuit}^{\mathbb{F}(1)}$. Let $\operatorname{Fr}$ denote the Frobenius morphism for $H_{n}\times H_{n-1,\diamondsuit}$. Namely, consider the completed Rees sheaf $\mathcal{D}_{n,n-1,\hbar}^{\mathbb{F}}$, this is a coherent $\mathcal{D}_{n-1,\diamondsuit,\hbar}^{\mathbb{F}}$-$\mathcal{D}_{n,\hbar}^{\mathbb{F}}$-bimodule (and a sheaf in the Zariski topology on $H_{n}^{\mathbb{F}}\times H_{n-1,\diamondsuit}^{\mathbb{F}}$). Note that $\operatorname{Fr}_{*}\left(\mathcal{D}_{n-1,\diamondsuit,\hbar}^{\mathbb{F}}\widehat{\otimes}_{\mathbb{F}[[\hbar]]}\mathcal{D}_{n,\hbar}^{\mathbb{F},opp}\right)$ is a coherent sheaf of $\mathcal{O}_{H_{n}^{\mathbb{F}(1)}\times H_{n-1,\diamondsuit}^{\mathbb{F}(1)}}[[\hbar]]$-modules. It follows that $\operatorname{Fr}_{*}(\mathcal{D}_{n,n-1,\hbar}^{\mathbb{F}})$ is also a coherent sheaf of $\mathcal{O}_{H_{n}^{\mathbb{F}(1)}\times H_{n-1,\diamondsuit}^{\mathbb{F}(1)}}[[\hbar]]$-modules.

Now note that, by the construction, $\operatorname{Fr}_{*}(\mathcal{D}_{n,n-1,\hbar}^{\mathbb{F}})$ is $T_{c}^{\mathbb{F}}$-equivariant. Since the action of $T_{c}^{\mathbb{F}}$-contracting, the functor of $\hbar$-adic completion is a category equivalence

[TABLE]

Let $\underline{\mathcal{D}}_{n,n-1,\hbar}^{\mathbb{F}}$ denote the $T_{c}^{\mathbb{F}}$-equivariant coherent sheaf on $H_{n}^{\mathbb{F}(1)}\times H_{n-1,\diamondsuit}^{\mathbb{F}(1)}\times\mathbb{A}^{1}$ corresponding to $\operatorname{Fr}_{*}(\mathcal{D}_{n,n-1,\hbar}^{\mathbb{F}})$. Set $\underline{\mathcal{D}}_{n,n-1}^{\mathbb{F}}:=\underline{\mathcal{D}}_{n,n-1,\hbar}^{\mathbb{F}}/(\hbar-1)$. This is a coherent sheaf on $H_{n}^{\mathbb{F}(1)}\times H_{n-1,\diamondsuit}^{\mathbb{F}(1)}$. Note that its restriction to the conical topology admits a natural embedding into $\operatorname{Fr}_{*}\mathcal{D}^{\mathbb{F}}_{n,n-1}$.

Let us record some basic properties of $\underline{\mathcal{D}}_{n,n-1}^{\mathbb{F}},\underline{\mathcal{D}}_{n,n-1,\hbar}^{\mathbb{F}}$.

Lemma 4.7**.**

The following claims are true:

1. (1)

The sheaf $\underline{\mathcal{D}}_{n,n-1,\hbar}^{\mathbb{F}}$ is flat over $\mathbb{A}^{1}$ and the specialization to $\hbar=0$ equals $\operatorname{Fr}_{*}\mathcal{O}^{\mathbb{F}}_{n,n-1}$. 2. (2)

The sheaf $\underline{\mathcal{D}}_{n,n-1}^{\mathbb{F}}$ carries a natural structure of a $\underline{\mathcal{D}}_{n-1,\diamondsuit}^{\mathbb{F}}$-$\mathcal{D}_{n}^{\mathbb{F}}$-bimodule. 3. (3)

We have $R\Gamma(\underline{\mathcal{D}}_{n-1,\diamondsuit}^{\mathbb{F}})\cong\mathcal{A}_{n-1,\diamondsuit}^{\mathbb{F}}$, an isomorphism of $\mathcal{A}_{n-1,\diamondsuit}^{\mathbb{F}}$-$\mathcal{A}_{n}^{\mathbb{F}}$-bimodules.

Proof.

(1) follows directly from the constructions of $\underline{\mathcal{D}}^{\mathbb{F}}_{n,n-1,\hbar},\underline{\mathcal{D}}^{\mathbb{F}}_{n,n-1}$. To prove (2), notice that the construction that produces $\underline{\mathcal{D}}^{\mathbb{F}}_{n,n-1}$ from $\mathcal{D}^{\mathbb{F}}_{n,n-1}$ also applies to $\mathcal{D}^{\mathbb{F}}_{n},\mathcal{D}^{\mathbb{F}}_{n-1,\diamondsuit}$ and produces $\underline{\mathcal{D}}^{\mathbb{F}}_{n},\underline{\mathcal{D}}^{\mathbb{F}}_{n-1,\diamondsuit}$. (3) follows from the construction of $\underline{\mathcal{D}}^{\mathbb{F}}_{n,n-1}$ and the analogous properties of $\mathcal{D}^{\mathbb{F}}_{n,n-1}$ in Lemma 4.6. ∎

It turns out that $\underline{\mathcal{D}}^{\mathbb{F}}_{n,n-1}$ is a “Frobenius constant” quantization of $\mathcal{O}_{n,n-1}^{\mathbb{F}}$ in the sense of the following lemma.

Lemma 4.8**.**

The action of $\mathcal{O}_{H_{n}^{\mathbb{F}(1)}\times H_{n-1}^{\diamondsuit\mathbb{F}(1)}}$ on $\underline{\mathcal{D}}^{\mathbb{F}}_{n,n-1}$ factors through $\mathcal{O}_{n,n-1}^{\mathbb{F}(1)}$.

Proof.

Since $\underline{\mathcal{D}}^{\mathbb{F}}_{n,n-1}$ embeds into $\operatorname{Fr}_{*}\mathcal{D}_{n,n-1}^{\mathbb{F}}$, it is enough to prove our claim for the latter sheaf.

Let $\iota$ denote the inclusion $V^{0\mathbb{F}(1)}/S_{n}\times V^{0\mathbb{F}(1)}/S_{n-1}\hookrightarrow H_{n}^{\mathbb{F}(1)}\times H_{n-1,\diamondsuit}^{\mathbb{F}(1)}$. Note that the pull-back $\iota^{*}\operatorname{Fr}_{*}\mathcal{D}^{\mathbb{F}}_{n,n-1}$ coincides with the microlocalization of the bimodule $\Gamma(\mathcal{D}_{n,n-1}^{\mathbb{F}})=\mathcal{A}_{n-1,\diamondsuit}^{\mathbb{F}}$ to $V_{n}^{0\mathbb{F}(1)}/S_{n}\times V^{0\mathbb{F}(1)}/S_{n-1}$ (that can also be viewed as an open subset of $V_{n}^{\mathbb{F}(1)}/S_{n}\times V_{n}^{\mathbb{F}(1)}/S_{n-1}$). It is easy to see that the $\mathbb{F}[V_{n}^{\mathbb{F}(1)}]^{S_{n-1}}$-$\mathbb{F}[V_{n}^{\mathbb{F}(1)}]^{S_{n}}$-bimodule $\mathcal{A}_{n-1,\diamondsuit}^{\mathbb{F}}$ is scheme theoretically supported on the diagonal $V_{n}^{\mathbb{F}(1)}/S_{n-1}\hookrightarrow V_{n}^{\mathbb{F}(1)}/S_{n}\times V_{n}^{\mathbb{F}(1)}/S_{n-1}$. Hence $\iota^{*}\operatorname{Fr}_{*}\mathcal{D}^{\mathbb{F}}_{n,n-1}$ is scheme theoretically supported on $H_{n,n-1}^{\mathbb{F}(1)}\cap(V_{n}^{0\mathbb{F}(1)}/S_{n}\times V_{n}^{0\mathbb{F}(1)}/S_{n-1})$.

Now note that $\operatorname{Fr}_{*}\mathcal{O}_{H_{n,n-1}^{\mathbb{F}}}\subset\iota_{*}\iota^{*}\operatorname{Fr}_{*}\mathcal{O}_{H_{n,n-1}^{\mathbb{F}}}.$ Since $\operatorname{Fr}_{*}\mathcal{D}^{\mathbb{F}}_{n,n-1}$ is a microlocal filtered quantization of $\operatorname{Fr}_{*}\mathcal{O}_{H_{n,n-1}^{\mathbb{F}}}$, it follows that

[TABLE]

Since $\iota^{*}\operatorname{Fr}_{*}\mathcal{D}^{\mathbb{F}}_{n,n-1}$ is scheme theoretically supported on $H_{n,n-1}^{(1)\mathbb{F}}\cap(V_{n}^{0\mathbb{F}(1)}/S_{n}\times V_{n}^{0\mathbb{F}(1)}/S_{n-1})$, we deduce that $\operatorname{Fr}_{*}\mathcal{D}_{n,n-1}^{\mathbb{F}}$ is scheme theoretically supported on $H_{n,n-1}^{\mathbb{F}(1)}$, which is what we need to prove. ∎

Here is the final property of $\underline{\mathcal{D}}_{n,n-1}^{\mathbb{F}}$ we need.

Lemma 4.9**.**

The sheaf $\underline{\mathcal{D}}^{\mathbb{F}}_{n,n-1}$ on $H_{n,n-1}^{\mathbb{F}(1)}$ is a splitting bundle for the Azumaya algebra

[TABLE]

Proof.

As was mentioned in Section 3.1, the rank of the Azumaya algebra $\underline{\mathcal{D}}^{\mathbb{F}}_{n}$ is equal to $p^{2n}$. Similarly, the rank of $\underline{\mathcal{D}}^{\mathbb{F}}_{n-1,\diamondsuit}$ By (2) of Lemma 4.7, $\underline{\mathcal{D}}_{n,n-1}^{\mathbb{F}}$ is a module over (4.2). So the rank of every fiber of $\underline{\mathcal{D}}_{n,n-1}^{\mathbb{F}}$ is $\geqslant p^{2n}$. What we need to prove is that the rank of every fiber is $p^{2n}$. It is enough to prove the analogous claim for $\underline{\mathcal{D}}_{n,n-1,\hbar}^{\mathbb{F}}$ viewed as a coherent sheaf on $H_{n,n-1}^{\mathbb{F}(1)}\times\mathbb{A}^{1}$. Recall that this sheaf is $T_{c}^{\mathbb{F}}$-equivariant and that the action is contracting. By (1) of Lemma 4.7, the rank of $\underline{\mathcal{D}}^{\mathbb{F}}_{n,n-1,\hbar}$ on $H_{n,n-1}^{\mathbb{F}(1)}\times\{0\}$ is $p^{2n}$. By the semi-continuity of rank, the rank at any other point of $H_{n,n-1}^{\mathbb{F}(1)}\times\mathbb{A}^{1}$ is $\leqslant p^{2n}$. This completes the proof. ∎

5. Proof of (1.4) in characteristic $p\gg 0$

The goal of this section is to establish an $\tilde{A}_{n-1,\diamondsuit}$-linear isomorphism $\alpha_{n*}\beta_{n}^{*}\mathcal{P}_{n-1,\diamondsuit}^{\mathbb{F}}\xrightarrow{\sim}\mathcal{P}_{n}^{\mathbb{F}}$. The proof is in two steps. First, we use results of Section 4.3 together with the construction of the Procesi bundles from [BK2] recalled in Section 3.2 to prove a weaker version of (1.4), where we twist by some line bundles. Then we prove that no nontrivial twists can occur.

5.1. Weaker version of (1.4)

Our goal in this section is to prove the following statement. We use the notation from Section 3.2.

Proposition 5.1**.**

There are integers $k,\ell$ such that there exists an $\widetilde{A}_{n-1,\diamondsuit}^{\mathbb{F}}$-linear isomorphism

[TABLE]

Proof.

Step 1. First of all, we can replace $H_{n},V_{n}$ etc. with their Frobenius twists. We claim that it is also enough to prove the completed version of (5.1), where we replace $H_{n}^{\mathbb{F}(1)}$ with $H_{n}^{\mathbb{F}(1)\wedge}:=\rho^{-1}(V_{n}^{\mathbb{F}(1)\wedge}/S_{n})$, etc.. So suppose we know $\alpha_{n*}\beta_{n}^{*}\hat{\mathcal{P}}_{n-1,\diamondsuit}^{\mathbb{F}}(k)\cong\hat{\mathcal{P}}_{n}^{\mathbb{F}}(\ell)$ and we want to deduce (5.1).

Recall from Section 3.2 that the bundle $\hat{\mathcal{P}}_{n}^{\mathbb{F}}$ is rigid and admits a $T_{c}^{\mathbb{F}(1)}$-equivariant structure. As was mentioned in Section 3.2, each equivariant structure gives rise to an extension of $\hat{\mathcal{P}}_{n}^{\mathbb{F}}$ to $H_{n}^{\mathbb{F}(1)}$ but the extension as a vector bundle is independent of the choice of an equivariant structure. Now take a $T_{c}^{\mathbb{F}(1)}$-equivariant structure on $\hat{\mathcal{P}}_{n-1,\diamondsuit}^{\mathbb{F}}$. It gives rise to an equivariant structure on $\hat{\mathcal{P}}_{n-1,\diamondsuit}^{\mathbb{F}}(k)$ and, since $\alpha_{n},\beta_{n}$ are equivariant, also on $\hat{\mathcal{P}}_{n}^{\mathbb{F}}(\ell)$. The latter comes from the isomorphism $\hat{\mathcal{P}}_{n}^{\mathbb{F}}(\ell)\cong\alpha_{n*}\circ\beta_{n}^{*}(\hat{\mathcal{P}}_{n-1,\diamondsuit}^{\mathbb{F}}(k))$. This equivariant structure on $\hat{\mathcal{P}}_{n}^{\mathbb{F}}(\ell)$ will give rise to (5.1).

Step 2. Consider the following diagram.

[TABLE]

Here $\operatorname{Res}$ is the restriction functor for the inclusion $\mathcal{A}^{\mathbb{F}}_{n}\hookrightarrow\mathcal{A}^{\mathbb{F}}_{n-1,\diamondsuit}$. Note that the vertical arrows are equivalences by [BK2, Proposition 2.2]. Also note that the diagram is commutative. Indeed, the composition

[TABLE]

is the derived tensor product with $R\Gamma(\underline{\mathcal{D}}^{\mathbb{F}}_{n,n-1})$. By (3) of Lemma 4.7, the latter object is $\mathcal{A}_{n-1,\diamondsuit}^{\mathbb{F}}$. The commutativity follows.

(5.2) remains commutative after we restrict it to $V_{n}^{\mathbb{F}(1)\wedge}$. So we get the following commutative diagram.

[TABLE]

Step 3. Consider the vector bundle $\underline{\mathcal{D}}_{n,n-1}^{\mathbb{F}\wedge}$ on $H_{n,n-1}^{\mathbb{F}(1)\wedge}$. By Lemma 4.9, it is a splitting bundle for the Azumaya algebra $\alpha_{n}^{(1)*}\underline{\mathcal{D}}_{n}^{\mathbb{F}\wedge,opp}\otimes\beta_{n}^{(1)*}\underline{\mathcal{D}}_{n-1,\diamondsuit}^{\mathbb{F}\wedge}$. But both Azumaya algebras $\underline{\mathcal{D}}_{n}^{\mathbb{F}\wedge,opp}$ and $\underline{\mathcal{D}}_{n-1,\diamondsuit}^{\mathbb{F}\wedge}$ split with splitting bundles $\hat{\mathcal{E}}_{n}^{\mathbb{F}*},\hat{\mathcal{E}}_{n-1,\diamondsuit}^{\mathbb{F}}$, respectively. Since a splitting bundle is defined uniquely up to a twist with a line bundle, there is a line bundle $\mathcal{L}$ on $H_{n,n-1}^{\mathbb{F}(1)\wedge}$ such that we have the following isomorphism:

[TABLE]

We claim that $\mathcal{L}=\mathcal{O}(-k,\ell)^{\wedge}$ for some $k,\ell\in\mathbb{Z}$. Indeed, the higher cohomology of $\mathcal{O}_{H_{n,n-1}^{\mathbb{F}(1)\wedge}}$ vanishes because the same holds for $H_{n,n-1}^{\mathbb{F}}$. So every line bundle on $H_{n,n-1}^{\mathbb{F}(1)\wedge}$ is rigid. Therefore it extends to $H_{n,n-1}^{\mathbb{F}(1)}$. Our claim that $\mathcal{L}=\mathcal{O}(-k,\ell)^{\wedge}$ follows from Lemma 2.9. So we see that

[TABLE]

Step 4. The following commutative diagram is a consequence of (5.5) and commutative diagram (5.3):

[TABLE]

Step 5. Let $\hat{\mathcal{B}}^{\mathbb{F}}_{n},\hat{\mathcal{B}}^{o,\mathbb{F}}_{n}$ have the same meaning as in Section and let $\hat{\mathcal{B}}_{n-1,\diamondsuit}^{\mathbb{F}},\hat{\mathcal{B}}_{n-1,\diamondsuit}^{o,\mathbb{F}}$ be defined similarly. We have

[TABLE]

The first two lines are from Section 3.2, while the third line follows from the first two and (5.5).

We claim that we have the following isomorphism of $\widetilde{A}_{n-1,\diamondsuit}^{\mathbb{F}(1)\wedge}$-$\widetilde{A}_{n}^{\mathbb{F}(1)\wedge}$-bimodules

[TABLE]

Indeed, recall from Section 3.2 that the Morita equivalence between $\mathcal{A}_{n}^{\mathbb{F}\wedge}$ and $\widetilde{A}_{n}^{\mathbb{F}(1)\wedge}$ is composed of two Morita equivalences:

• (a)

$\mathsf{W}(V_{n}^{\mathbb{F}})^{\wedge}\otimes_{\mathcal{A}_{n}^{\mathbb{F}\wedge}}\bullet:\mathcal{A}_{n}^{\mathbb{F}\wedge}\operatorname{-mod}\xrightarrow{\sim}\mathsf{W}(V_{n}^{\mathbb{F}})^{\wedge}\operatorname{-mod}^{S_{n}}$,

• (b)

the Morita equivalence $\mathsf{W}(V_{n}^{\mathbb{F}})^{\wedge}\operatorname{-mod}^{S_{n}}\xrightarrow{\sim}\mathbb{F}[V_{n}^{\mathbb{F}(1)}]^{\wedge}\operatorname{-mod}^{S_{n}}$ given by the standard splitting module $\mathbb{F}[\underline{x}^{p}][[\underline{y}]]$ for $\mathsf{W}(V_{n}^{\mathbb{F}})^{\wedge}$, which is $S_{n}$-equivariant.

The Morita equivalence for $n-1$ is defined similarly, where in (b) we use the same standard splitting module. So applying the equivalences in (a) on the left and on the right to $\mathcal{A}_{n-1,\diamondsuit}^{\mathbb{F}\wedge}$ we get

[TABLE]

And then applying the equivalences in (b), we get $\widetilde{A}_{n}^{\mathbb{F}(1)\wedge}$. This proves (5.7).

Step 6. Applying the Morita equivalences from Step 5 to (5.6)

[TABLE]

As in Step 2, the vertical arrows are equivalences. Computing the image of $\widetilde{A}^{\mathbb{F}(1)\wedge}_{n-1,\diamondsuit}$ in $D^{b}(\operatorname{Coh}(H^{\mathbb{F}(1)\wedge}_{n}))$ using diagram (5.8) in two different ways we arrive at $\alpha_{n*}\beta_{n}^{*}(\hat{\mathcal{P}}^{\mathbb{F}}_{n-1,\diamondsuit}(\ell))\cong\hat{\mathcal{P}}^{\mathbb{F}}_{n}(k)$, an $\widetilde{A}_{n-1,\diamondsuit}^{\mathbb{F}(1)\wedge}$-linear isomorphism. Thanks to Step 1, the proof of the proposition is complete. ∎

5.2. Absence of twists

Here we prove the following lemma.

Lemma 5.2**.**

In Proposition 5.1 we have $k=\ell=0$.

Proof.

Recall that $(\mathcal{P}^{\mathbb{F}}_{n-1,\diamondsuit})^{S_{n-1}}\cong\mathcal{O}_{n-1,\diamondsuit}^{\mathbb{F}}$ by the construction. So (5.1) implies

[TABLE]

Consider the restrictions of both sides of (5.9) to $\rho_{n}^{-1}((V_{n}^{\mathbb{F}}/S_{n})^{\wedge_{b}})$, where $b$ is as in Lemma 2.3.

By Proposition 3.3, the right hand side of (5.9) is either $\mathcal{T}_{n}^{\mathbb{F}}(k)$ or $\mathcal{T}_{n}^{\mathbb{F}^{*}}(k)$. So by Lemma 2.14, the restriction of the right hand side is

[TABLE]

where the plus sign corresponds to $\mathcal{T}_{n}^{\mathbb{F}}$ and the minus sign corresponds to $\mathcal{T}_{n}^{\mathbb{F}*}$.

The restriction of $\mathcal{O}_{n,n-1}^{\mathbb{F}}(\ell,0)$ to $\alpha_{n}^{-1}(\rho^{-1}((V_{n}^{\mathbb{F}}/S_{n})^{\wedge_{b}}))$ is $\mathcal{O}^{\mathbb{F}\wedge}_{n,n-1}$ for the components of form (i) (in Lemma 2.8) and $\mathcal{O}^{\mathbb{F}\wedge}_{n,n-1}(\ell)$ for the component of form (ii). Let us compute $\alpha_{2*}(\mathcal{O}^{\mathbb{F}\wedge}_{2,1}(\ell))$. First, note that $\alpha^{*}_{2}\mathcal{O}_{2}^{\mathbb{F}}(1)=\mathcal{O}_{2,1}^{\mathbb{F}}(2)$. Thanks to the projection formula, it is enough to compute $\alpha_{2*}(\mathcal{O}^{\mathbb{F}\wedge}_{2,1}(\ell))$ for $\ell=0,-1$. We claim that here $\alpha_{2*}\mathcal{O}_{2,1}^{\mathbb{F}}(\ell)\cong\mathcal{O}_{2}^{\mathbb{F}}(\ell)\oplus\mathcal{O}_{2}^{\mathbb{F}}(\ell+1)$. Indeed, we already know this for $\ell=0$, while the bundle $\mathcal{O}_{2,1}^{\mathbb{F}}(-1)$ is the canonical bundle of $H^{\mathbb{F}}_{n,n-1}$ so $\alpha_{2*}\mathcal{O}_{2,1}^{\mathbb{F}}(-1)=(\alpha_{2*}\mathcal{O}_{2,1}(0))^{*}$.

We conclude that, for $\ell=2\ell_{1}+\ell_{0}$ with $\ell_{0}\in\{-1,0\}$, the restriction of the left hand side of (5.9) is

[TABLE]

Comparing (5.10) and (5.11), we see that we have the following options:

• (I)

$k=\ell=0$.

• (II)

$k=0,\ell=-1$.

• (III)

$n=3,k=\ell=1$.

In particular, $\mathcal{P}_{2}^{\mathbb{F}}=\mathcal{O}^{\mathbb{F}}\oplus\mathcal{O}^{\mathbb{F}}(1)$ (here $\ell=0$ by default).

Let us show that (III) is not possible. Here we have $(\mathcal{P}_{3}^{\mathbb{F}})^{S_{2}}\cong\mathcal{T}_{3}^{\mathbb{F}}$. From Lemma 3.4 it follows that $\mathcal{O}_{3}^{\mathbb{F}}(2)=\left(\mathcal{P}_{3}^{\mathbb{F}}(1)\right)^{\operatorname{sgn}}$. Note that

[TABLE]

By the computations above, the restriction of the left hand side to $\rho^{-1}((V_{3}^{\mathbb{F}}/S_{3})^{\wedge_{b}})$ is $\mathcal{O}^{\mathbb{F}}(0)\oplus\mathcal{O}^{\mathbb{F}}(1)\oplus\mathcal{O}^{\mathbb{F}}(2)$. However, the restriction of $(\mathcal{P}_{3}^{\mathbb{F}}(1))^{S_{2},\operatorname{sgn}}$ may only have two pairwise nonisomorphic summands, compare to the proof of Lemma 3.4.

Let us show that (II) is not possible. We induct on $n$ with the base of $n=2$ done above. In particular, $(\mathcal{P}_{n-1}^{\mathbb{F}})^{S_{n-2}}=\mathcal{T}_{n-1}^{\mathbb{F}}$ and hence, by Lemma 3.4, $(\mathcal{P}_{n-1}^{\mathbb{F}})^{\operatorname{sgn}}=\mathcal{O}^{\mathbb{F}}(1)$. On the other hand, $(\mathcal{P}^{\mathbb{F}}_{n})^{S_{n-1}}=\mathcal{T}_{n}^{\mathbb{F}*}$. From (5.1) we deduce that

[TABLE]

But $(\mathcal{P}^{\mathbb{F}}_{n})^{S_{n-1},\operatorname{sgn}}=(\mathcal{P}^{\mathbb{F}}_{n})^{\operatorname{sgn}}\otimes\left((\mathcal{P}^{\mathbb{F}}_{n})^{S_{n-1}}\right)^{*}=\mathcal{T}_{n}^{\mathbb{F}}(-1)$, the former equality follows from Corollary 3.5, while the latter holds thanks to Lemma 3.4. Restricting both sides of (5.12) to $\rho_{n}^{-1}((V_{n}^{\mathbb{F}}/S_{n})^{\wedge_{b}})$ we see that the restrictions differ by a twist with $\mathcal{O}(-1)$, which leads to a contradiction. ∎

6. On normalization of $IH_{n}$

6.1. Theorem 1.3 in characteristic $p$

Proposition 6.1**.**

The following claims are true:

1. (1)

The sheaf $\mathcal{P}_{n}^{\mathbb{F}}$ has a natural algebra structure, let $X_{n}^{\mathbb{F}}$ denote the relative spectrum of $\mathcal{P}_{n}^{\mathbb{F}}$. 2. (2)

The variety $X_{n}^{\mathbb{F}}$ is normal, Cohen-Macaulay and Gorenstein. The canonical bundle is obtained by pulling back $\mathcal{O}^{\mathbb{F}}_{n}(-1)$ from $H_{n}^{\mathbb{F}}$. 3. (3)

The natural morphism $X_{n}^{\mathbb{F}}\rightarrow IH_{n}^{\mathbb{F}}$ is bijective and is an isomorphism over $V_{n}^{1\mathbb{F}}/S_{n}$.

Proof.

We will prove this by induction on $n$ starting with $n=2$. Here $IH^{\mathbb{F}}_{2}=U^{\mathbb{F}}_{2}=H^{\mathbb{F}}_{2,1}$ are Cohen-Macaulay and Gorenstein and (1)-(3) hold. Now suppose that they hold for $n-1$ and prove them for $n$.

Step 1. Let us construct a sheaf of algebras structure on $\mathcal{P}_{n}^{\mathbb{F}}$. Set $X^{\mathbb{F}}_{n,n-1}:=H^{\mathbb{F}}_{n,n-1}\times_{H^{\mathbb{F}}_{n-1}}X^{\mathbb{F}}_{n-1}$. This scheme is finite and flat over $H^{\mathbb{F}}_{n,n-1}$ so it is Cohen-Macaulay. Over $V_{n-1,\diamondsuit}^{0\mathbb{F}}/S_{n-1}\subset H_{n,n-1}^{\mathbb{F}}$, the scheme $X^{\mathbb{F}}_{n,n-1}$ becomes $V^{0,\mathbb{F}}_{n-1,\diamondsuit}$. Therefore $X^{\mathbb{F}}_{n,n-1}$ is generically reduced, hence reduced. We denote the natural morphism $X^{\mathbb{F}}_{n,n-1}\rightarrow H^{\mathbb{F}}_{n,n-1}$ by $\pi_{n,n-1}$ and the morphism $X^{\mathbb{F}}_{n,n-1}\rightarrow X^{\mathbb{F}}_{n-1,\diamondsuit}$ by $\widetilde{\beta}_{n}$.

Consider the composition $\alpha_{n}\circ\pi_{n,n-1}:X^{\mathbb{F}}_{n,n-1}\rightarrow H^{\mathbb{F}}_{n}$. Note that $(\alpha_{n}\circ\pi_{n,n-1})_{*}\mathcal{O}_{X^{\mathbb{F}}_{n,n-1}}=\alpha_{n*}\beta_{n}^{*}\mathcal{P}_{n-1,\diamondsuit}^{\mathbb{F}}$. Hence, by Proposition 5.1 combined with Lemma 5.2,

[TABLE]

an $\widetilde{A}_{n-1,\diamondsuit}^{\mathbb{F}}$-linear isomorphism. This isomorphism, in particular, equips $\mathcal{P}_{n}^{\mathbb{F}}$ with the structure of a sheaf of algebras.

Step 2. Set $X_{n}^{\mathbb{F}}:=\operatorname{Spec}_{\mathcal{O}_{n}^{\mathbb{F}}}(\mathcal{P}_{n}^{\mathbb{F}})$. By the construction of $X_{n}^{\mathbb{F}}$ we have a natural morphism $\widetilde{\alpha}_{n}:X_{n,n-1}^{\mathbb{F}}\rightarrow X_{n}^{\mathbb{F}}$. It satisfies

[TABLE]

So we have a degree $n!$ finite morphism $X_{n}^{\mathbb{F}}\rightarrow H_{n}^{\mathbb{F}}$ to be denoted by $\pi_{n}$, it satisfies $\alpha_{n}\circ\pi_{n,n-1}=\pi_{n}\circ\widetilde{\alpha}_{n}$. In particular, $X_{n}^{\mathbb{F}}$ is Cohen-Macaulay. Over $V_{n}^{0,\mathbb{F}}/S_{n}\subset H_{n}^{\mathbb{F}}$, the morphism $\alpha_{n}\circ\pi_{n,n-1}$ becomes the natural morphism $V^{0\mathbb{F}}_{n}\rightarrow V^{0\mathbb{F}}_{n}/S_{n}$. Therefore $X_{n}^{\mathbb{F}}\times_{H_{n}}(V_{n}^{0\mathbb{F}}/S_{n})=V_{n}^{0\mathbb{F}}$. It follows that $X_{n}^{\mathbb{F}}$ is reduced.

Step 3. Let us construct a morphism $X_{n}^{\mathbb{F}}\rightarrow IH_{n}^{\mathbb{F}}$. Note that $X_{n,n-1}^{\mathbb{F}}$ comes with a natural morphism $\widetilde{\rho}_{n,n-1}:X_{n,n-1}^{\mathbb{F}}\rightarrow V_{n}^{\mathbb{F}}$. (6.2) shows that $\widetilde{\alpha}_{n}$ identifies $\mathbb{F}[X_{n,n-1}]$ and $\mathbb{F}[X_{n}]$. This gives rise to $\widetilde{\rho}_{n}:X_{n}^{\mathbb{F}}\rightarrow V_{n}^{\mathbb{F}}$ with $\widetilde{\rho}_{n,n-1}=\widetilde{\rho}_{n}\circ\widetilde{\alpha}_{n}$. The compositions $X_{n}^{\mathbb{F}}\rightarrow V_{n}^{\mathbb{F}}\rightarrow V_{n}^{\mathbb{F}}/S_{n}$ and $X_{n}^{\mathbb{F}}\rightarrow H_{n}^{\mathbb{F}}\rightarrow V_{n}^{\mathbb{F}}/S_{n}$ coincide over $V_{n}^{0\mathbb{F}}/S_{n}$, hence coincide. Therefore we get a morphism $\iota_{n}:X_{n}^{\mathbb{F}}\rightarrow IH_{n}^{\mathbb{F}}$. This morphism is finite by construction and is an isomorphism over $V_{n}^{0\mathbb{F}}\subset IH_{n}^{\mathbb{F}}$.

Step 4. By [H1, Lemma 3.3.1] and the case of $n=2$, $IH_{n}^{\mathbb{F}}$ is smooth over $V_{n}^{1\mathbb{F}}/S_{n}$. It follows that $\iota_{n}$ is an isomorphism over $V_{n}^{1\mathbb{F}}/S_{n}$. In particular, $X_{n}^{\mathbb{F}}$ is smooth outside a codimension $2$ locus. Since $X_{n}^{\mathbb{F}}$ is Cohen-Macaulay, it is normal and $\iota_{n}$ is the normalization morphism.

Step 5. Let us show that $X_{n}^{\mathbb{F}}$ is Gorenstein with canonical bundle $\pi_{n}^{*}\mathcal{O}_{n}^{\mathbb{F}}(-1)$. As in [H3, Section 6.1], we use the inductive assumption – $K_{X_{n-1}^{\mathbb{F}}}=\pi_{n-1}^{*}(\mathcal{O}_{n-1}^{\mathbb{F}}(-1))$ – to see that the canonical bundle of $X_{n,n-1}^{\mathbb{F}}$ is $\pi_{n,n-1}^{*}(\mathcal{O}_{n,n-1}^{\mathbb{F}}(0,-1))$. Therefore $\pi_{n}^{*}(\mathcal{O}_{n}^{\mathbb{F}}(-1))=\tilde{\alpha}_{n*}\pi_{n,n-1}^{*}(\mathcal{O}_{n,n-1}^{\mathbb{F}}(0,-1))$ is the canonical sheaf of $X_{n}^{\mathbb{F}}$.

Step 6. It remains to show that $\iota_{n}:X_{n}^{\mathbb{F}}\rightarrow IH_{n}^{\mathbb{F}}$ is bijective. By the inductive assumption, $\iota_{n-1}:X_{n-1}^{\mathbb{F}}\rightarrow IH_{n-1}^{\mathbb{F}}$ is bijective. Therefore the natural morphism $\iota_{n,n-1}:X_{n,n-1}^{\mathbb{F}}\rightarrow IH_{n,n-1}^{\mathbb{F}}$ is bijective as well. The fibers of the natural morphism $\hat{\alpha}_{n}:IH^{\mathbb{F}}_{n,n-1}\rightarrow IH^{\mathbb{F}}_{n}$ (as algebraic varieties) coincide with the fibers of $\bar{\alpha}_{n}:H^{\mathbb{F}}_{n,n-1}\rightarrow U^{\mathbb{F}}_{n}$. The latter fibers are all connected. So we see that the fibers of $\iota_{n}\circ\widetilde{\alpha}_{n}:X_{n,n-1}^{\mathbb{F}}\rightarrow IH_{n}^{\mathbb{F}}$ are connected. It follows that the finite morphism $\iota_{n}$ is bijective. ∎

6.2. Proof of main results

In this section we prove Theorems 1.3 and 1.2.

Consider the morphism $IH_{n}^{\mathsf{R}}\times_{V_{n}^{\mathsf{R}}/S_{n}}V_{n}^{1\mathsf{R}}/S_{n}\rightarrow H_{n}^{\mathsf{R}}$. It is quasi-finite so canonically decomposes as $IH_{n}^{\mathsf{R}}\times_{V_{n}^{\mathsf{R}}/S_{n}}V_{n}^{1\mathsf{R}}/S_{n}\rightarrow X_{n}^{\mathsf{R}}\rightarrow H_{n}^{\mathsf{R}}$ for a normal scheme $X_{n}^{\mathsf{R}}$, where the first arrow is an open embedding and the second arrow is finite. Thanks to Proposition 6.1, for $p$ sufficiently large, the base change of $X_{n}^{\mathsf{R}}$ to $\mathbb{F}$ is the scheme $X_{n}^{\mathbb{F}}$ from that proposition. Using Proposition 6.1 again, we see that $X_{n}^{\mathsf{R}}$ is Cohen-Macaulay after replacing $\mathsf{R}$ with a finite localization. Then we can define the bundle $\mathcal{P}_{n}^{\mathsf{R}}$ on $H_{n}^{\mathsf{R}}$ as $\pi_{n*}\mathcal{O}_{X_{n}^{\mathsf{R}}}$. By Proposition 6.1, it specializes to $\mathcal{P}_{n}^{\mathbb{F}}$ over $\mathbb{F}$. It follows that the specialization $\mathcal{P}_{n}$ to $\mathbb{C}$ has no higher cohomology and coincides with

[TABLE]

From the latter we see that it is a Procesi bundle. By the construction, it comes with a morphism $\mathcal{P}_{n}\rightarrow\alpha_{n*}\beta_{n}^{*}\mathcal{P}_{n-1,\diamondsuit}$ that must be an isomorphism. This proves Theorem 1.2. Then we can repeat the proof of the remaining parts of Proposition 6.1 over $\mathbb{C}$ to prove Theorem 1.3.