# Twisted equivariant HKR theorem for torus action and the small quantum group

Nicolas Hemelsoet

###### Abstract

We show that when a torus $T$ acts on a smooth variety $X$, the twisted HKR
isomorphism is equivariant. The main consequence is that the Bezrukavnikov-
Lachowska isomorphism, relating the Hochschild cohomology of the principal
block of the small quantum group to certain sheaf cohomology groups on the
Springer resolution $\widetilde{\mathcal{N}}$, can be upgraded to a ring
isomorphism by a twist.

###### Contents

  1. 1 Introduction
  2. 2 Geometry
    1. 2.1 Equivariant sheaves
    2. 2.2 Kontsevitch’s theorem
    3. 2.3 Equivariance of the twisted HKR isomorphism
  3. 3 Applications to the small quantum group
    1. 3.1 The small quantum group
    2. 3.2 Multiplicative structure of $HC[\tau]$

##  1 Introduction

The starting point of this article is the following theorem :

######  Theorem 1.1 ([1]).

There exists an algebra isomorphism

$HH^{\bullet}(\mathfrak{u}_{0}(\mathfrak{g}))\cong\bigoplus_{i+j+k=\bullet}H^{i}(\widetilde{\mathcal{N}},\wedge^{j}T\widetilde{\mathcal{N}})^{k}$

This theorem relates the Hochschild cohomology of the principal block of the
small quantum group with an ”equivariant” version of the Hochschild cohomology
of the Springer resolution. More formally, let us detail the notation :
$\mathfrak{g}$ is a semisimple Lie algebra, $\mathfrak{u}_{0}(\mathfrak{g})$
is the principal block of the small quantum group associated to
$\mathfrak{g}$, $\widetilde{\mathcal{N}}$ is the _Springer resolution_
$\widetilde{\mathcal{N}}:=T^{*}(G/B)=G\times_{B}\mathfrak{n}$ where $G$ is the
connected simple algebraic group of adjoint type associated to $\mathfrak{g}$,
$B\subset G$ is a Borel subgroup and
$\mathfrak{n}=[\mathfrak{b},\mathfrak{b}]$ where $\mathfrak{b}=Lie(B)$.
$HH^{\bullet}(\mathfrak{u}_{0}(\mathfrak{g}))$ is the Hochschild cohomology of
$\mathfrak{u}_{0}(\mathfrak{g})$. The subscript $k$ correspond to the grading
acquired from the $\mathbb{C}^{*}$ action by dilations on the fiber on
$\widetilde{\mathcal{N}}$. We refer to [1] and [12] for a more detailled
exposition.

The multiplication on the left-hand side is given by the Yoneda product, and
on the right-hand side by the exterior algebra structure. The purpose of this
paper is to explain precisely how to obtain a multiplicative isomorphism,
using a theorem by Kontsevitch. We will check the compatibility of the
multiplicative version of the Hochschild-Kostant-Rosenberg theorem with a
action of a torus.

Let $T$ be a complex torus, and $X$ be an irreducible smooth complex variety
acted on by $T$. More generally, one can consider an arbitrary complex
reductive group acting on a smooth variety $X$, if there exists an affine
cover by invariant open subset. Let $HH^{\bullet}(X)$ be the Hochschild
cohomology of $X$, defined as $Ext^{\bullet}_{Coh(X\times
X)}(\mathcal{O}_{\Delta},\mathcal{O}_{\Delta})$, where $\Delta\subset X\times
X$ is the diagonal (for a variety $Z$ we denote $\mathcal{O}_{Z}$ the
structure sheaf), and $HT^{\bullet}(X)$ denotes the bigraded vector space
$\bigoplus_{i,j}H^{i}(X,\wedge^{j}TX)$. Our first result is :

######  Theorem 1.2.

The twisted HKR morphism $HH^{\bullet}(X)\to HT^{\bullet}(X)$ is
$T$-equivariant.

This result relies on the following two propositions :

######  Proposition 1.3.

In the derived category of coherent sheaves on $X$ $D^{b}(\text{Coh}(X))$, the
quasi-isomorphism

$\iota^{*}\mathcal{O}_{\Delta}\cong\bigoplus_{i\in X}\Omega^{i}_{X}[i]$

is $T$-equivariant.

Here $\iota:\Delta\to X\times X$ is the inclusion map, $\Omega_{X}^{i}$ is the
sheaf of differential forms of degree $i$, and $[i]$ is the shift functor in
the derived category.

######  Proposition 1.4.

Let $t\in H\Omega^{\bullet}(X)$ be the Todd class of
$\widetilde{\mathcal{N}}$. Then, twisting with $t^{-1/2}$ gives a
$T$-equivariant multiplicative isomorphism $HH^{\bullet}(X)\cong
HT^{\bullet}(X)$.

When no group action is involved, the two statements are well-known ([7],
section $9$). The first proposition implies the famous Hochschild-Kostant-
Rosenberg (HKR) theorem relating the Hochschild cohomology of $X$ with sheaf
cohomology of poly-vector fields. The second statement relates the
multiplicative structures on both sides. Hence the essence of our results is
the compatibility with the action of $T$. Using our first theorem, we can
deduce an explicit way to make the Bezrukavnikov-Lachowska isomorphism
multiplicative. Setting $X=\widetilde{\mathcal{N}}$ and $T=\mathbb{C}^{*}$, we
will deduce in section $3$ the following theorem, describing geometrically the
multiplicative structure of $z_{0}(\mathfrak{g})$ :

######  Theorem 1.5.

The composition

$HH^{\bullet}(\mathfrak{u}_{0}(\mathfrak{g}))\xrightarrow{HKR}\bigoplus_{i+j+k=\bullet}H^{i}(\widetilde{\mathcal{N}},\wedge^{j}T\widetilde{\mathcal{N}})^{k}\xrightarrow{\langle-,Todd(\widetilde{\mathcal{N}})^{-1/2}\rangle}\bigoplus_{i+j+k=\bullet}H^{i}(\widetilde{\mathcal{N}},\wedge^{j}T\widetilde{\mathcal{N}})^{k}$

is a ring isomorphism.

Taking the zero degree part of the previous isomorphism gives the following
corollary:

######  Corollary 1.6.

The natural bigraded vector space isomorphism

$HH^{0}(\mathfrak{u}_{0}(\mathfrak{g}))\cong\bigoplus_{i+j+k=0}H^{i}(\widetilde{\mathcal{N}},\wedge^{j}T\widetilde{\mathcal{N}})^{k}$

can be upgraded to a multiplicative isomorphism using a twist.

The theorem 1.5 and the corollary 1.6 also hold for singular blocks, where one
needs to twist by the corresponding “parabolic Todd class”.

Let us briefly outline the contents of this paper : in section $2$ we study
equivariant sheaves on smooth varieties, give background material for the
Kontsevitch theorem and prove theorem 1.2. In section $3$ we explain the
connection with the small quantum group, and derive some consequences for the
structure of $z_{0}(\mathfrak{g})$.

### Acknowledgements

I would like to thanks Anna Lachowska and Qi You for support and suggesting
the project. I would also like to thanks Pieter Belmans, Damien Calaque,
Dmitriy Rumynin and Andras Szenes for helpful and interesting discussions.

##  2 Geometry

We first recall a theorem by Sumihiro :

######  Theorem 2.1.

[16] Let $T$ be a torus and $X$ a complex $T$-algebraic variety. Then, there
is an open cover $\mathfrak{U}$ of $X$, which is $T$-invariant (i.e for all
$U\in\mathfrak{U}$, we have $TU\subset U$).

This theorem reduces many statements about algebraic varieties with a
$T$-action to an affine statement about graded $R$-modules that are easier to
prove.

###  2.1 Equivariant sheaves

Let $X$ be a variety over the complex numbers, with an algebraic action of a
complex reductive group $K$ (which is not related to the Lie algebra
$\mathfrak{g}$ mentioned at the beginning of the introduction). We will use
$\rm{QCoh}^{K}(X)$ for the category of quasi-coherent $K$-equivariant sheaves.
First we recall the definitions and standard facts of an equivariant sheaf :

######  Definition 2.2.

A sheaf $\mathscr{F}\in\rm{QCoh}(X)$ is $K$-equivariant if there is an
isomorphism $\theta:m^{*}\mathscr{F}\cong pr_{2}^{*}\mathscr{F}$, where
$m,pr_{2}:K\times X\to X$ are respectively the action and the projection
(moreover, $\theta$ should satisfies a cocycle condition).

Taking the stalks of $\theta$ at $(g,x)\in K\times X$ gives an isomorphism
$\theta_{x}:\mathscr{F}_{g\cdot x}\cong\mathscr{F}_{x}$. A $K$-equivariant
sheaf $\mathscr{F}$ is equivalent to the data of a sheaf $\mathscr{F}$ with
such isomorphisms. We also have

######  Lemma 2.3.

A $K$-equivariant sheaf $\mathscr{F}$ is equivalent to a sheaf $\mathscr{F}$
with maps $g:\Gamma(U,\mathscr{F})\to\Gamma(g(U),\mathscr{F})$ for each $g\in
T$ and $U$ open, commuting with restrictions and such that
$g_{1}\cdot(g_{2}\cdot s)=(g_{1}\cdot g_{2})\cdot s$ for all $g_{1},g_{2}\in
K$ and all $s\in\mathscr{F}(U)$.

######  Corollary 2.4.

There is a natural structure of $K$-modules on $\Gamma(X,\mathscr{F})$ for any
equivariant sheaf $\mathscr{F}$. More generally, there is a structure of
$K$-modules on $Hom_{\rm{QCoh(X)}}(\mathscr{F}_{1},\mathscr{F}_{2})$ if
$\mathscr{F}_{i}\in\rm{QCoh}^{K}(X)$.

###### Proof.

The first point is clear. The action in general is defined as follow : if
$\varphi:\mathscr{F}_{1}(U)\to\mathscr{F}_{2}(U)$, we define
$g\cdot\varphi=g\circ\varphi\circ g^{-1}$, where $g$ denotes the map in lemma
2.3. ∎

######  Definition 2.5.

The category $\rm{QCoh}^{K}(X)$ is the category where objects are
$K$-equivariant quasi-coherent sheaves of $\mathcal{O}_{X}$-modules, and
morphisms are equivariant morphisms of $\mathcal{O}_{X}$-modules, i.e. the
maps $f:\mathscr{F}_{1}\to\mathscr{F}_{2}$ such that that $f(g\cdot s)=g\cdot
f(s)$ for all $s\in\mathscr{F}_{1}(U)$ and all $g\in T$.

To give more intuition, we recall that there is an equivalence of categories
between the category of locally free coherent $K$-equivariant sheaves and the
category of finite-dimensional vector bundles with an action of $K$, linear on
each fibers, where morphisms are $K$-equivariant morphisms of vector bundles
(see [5], chapter $5$).

######  Lemma 2.6.

The $K$-module $V:=Hom_{\rm{QCoh}(X)}(\mathscr{F}_{1},\mathscr{F}_{2})$ is
semisimple. Moreover there is an isomorphism
$Hom_{\rm{QCoh}^{K}(X)}(\mathscr{F}_{1},\mathscr{F}_{2})=(Hom_{\rm{QCoh}(X)}(\mathscr{F}_{1},\mathscr{F}_{2}))^{K}$.

###### Proof.

The induced $K$-action is algebraic, i.e given by a ring map $V\to
V\otimes\mathcal{O}(K)$. It follows easily that $V$ is the union of its
finite-dimensional $K$-modules, and hence semi-simple. The second part follows
from the definition of the $K$-action. ∎

We now investigate basic properties of $K$-equivariant sheaves in order to be
able to describe derived functors, now computed in the equivariant category.

######  Remark 2.7.

Let us emphasize that in geometric representation theory, the usual definition
of the ”equivariant derived category” (as defined e.g in Bernstein-Lunts :[2])
does not coincide with the derived category of equivariant sheaves (the one we
consider here).

######  Lemma 2.8.

For each $\mathscr{F}\in\rm{QCoh}^{K}(X)$, there is an injective object
$I\in\rm{QCoh}^{K}(X)$ and a $K$-equivariant monomorphism $\mathscr{F}\to I$.
Moreover $\text{Fgt}(I)\in\rm{QCoh(X)}$ is still injective, where
$\text{Fgt}:\rm{QCoh}^{K}(X)\to\rm{QCoh}(X)$ is the forgetful functor.

###### Proof.

The first part is proposition 5.1.2 in [8]. The second part follows from the
explicit construction of $I$ in the same paper. ∎

This lemma ensure that we can consider right derived functors of
$Hom(\mathscr{F}_{1},-)$.

######  Corollary 2.9.

Let $\mathscr{F}_{1}\in\rm{Coh(X)}^{K}(X)$ and
$\Gamma_{1}=Hom_{\rm{QCoh^{K}(X)}}(\mathscr{F}_{1},-):\rm{Coh^{K}(X)}\to
K-Mod$. Then there is a canonical isomorphism
$R(\Gamma_{1}\circ\text{Fgt})\cong R\Gamma_{1}\circ\text{Fgt}$.

######  Corollary 2.10.

For any equivariant sheaf $\mathscr{F}_{1}$, there is an isomorphism between
the functors $\rm{QCoh}^{K}(X)\to\rm{Vect}$ :

$R(\rm{Inv}\circ\Gamma_{\mathscr{F}_{1}})\cong\rm{Inv}\circ
R\Gamma_{\mathscr{F}_{1}}$

where
$\Gamma_{\mathscr{F}_{1}}:=Hom_{\rm{QCoh(X)}}(\text{Fgt}(\mathscr{F}_{1}),\text{Fgt}(-))$,
and $Inv:K-Mod\to\rm{Vect}$, $Inv(M)=M^{K}$. In particular if
$\mathscr{F}_{1},\mathscr{F}_{2}$ are $K$-equivariant sheaves on $X$, there is
an isomorphism

$Ext^{q}_{\rm{QCoh}^{K}(X)}(\mathscr{F}_{1},\mathscr{F}_{2})\cong(Ext^{q}_{\rm{QCoh}(X)}(\mathscr{F}_{1},\mathscr{F}_{2}))^{K}$

###### Proof.

Recall the Grothendieck formula to derive compatible left-exact functors :
$R(F\circ F^{\prime})\cong RF\circ RF^{\prime}$. Taking invariant is exact on
the subcategory of semisimple $K$-representations, and by lemma 2.6
$\Gamma(Hom_{\rm{QCoh(X)}}(\mathscr{F}_{1},\mathscr{F}_{2}))$ is semisimple.
Hence $R\text{Inv}=\text{Inv}$ and the formula follows. ∎

######  Corollary 2.11.

If $\mathscr{F}$ is a $K$-equivariant sheaf, then there is an isomorphism

$RHom_{\rm{QCoh^{K}(X)}}(\mathcal{O}_{X},\mathscr{F})\cong(RHom_{\rm{QCoh(X)}}(\mathcal{O}_{X},\mathscr{F}))^{K}$

######  Lemma 2.12.

If $0\to\mathscr{F}_{1}\to\mathscr{F}_{2}\to\mathscr{F}_{3}\to 0$ is a
$K$-equivariant sequence of $K$-equivariant sheaves, and $\mathscr{F}_{3}$ is
locally free, then the corresponding cohomology class in
$H^{1}(X,\mathscr{F}_{1}\otimes\mathscr{F}_{3}^{*})$ is $K$-invariant.

###### Proof.

By definition the short exact sequence gives a map
$\mathscr{F}_{3}\to\mathscr{F}_{1}[1]$ in $D^{b}(\rm{QCoh}^{K}(X))$. This is
the same as an element in
$Ext^{1}_{\rm{QCoh}^{K}(X)}(\mathscr{F}_{3},\mathscr{F}_{1})$. Since
$Ext^{1}_{\rm{QCoh}^{K}(X)}(\mathscr{F}_{3},\mathscr{F}_{1})\cong
Ext^{1}_{\rm{QCoh}^{K}(X)}(\mathcal{O}_{X},\mathscr{F}_{1}\otimes\mathscr{F}_{3}^{*})$
we can apply the corollary 2.10 and compute the latter as
$H^{1}(R\Gamma(X,\mathscr{F}_{1}\otimes\mathscr{F}_{3}^{*})^{K})$ which is
$K$-invariant. ∎

Now we specialize to the case where $K=T$ is a complex algebraic torus.

######  Lemma 2.13.

Assume that the $T$-action on $X$ is trivial and that $\mathcal{E}$ is a
$T$-equivariant vector bundle on $X$. Then, there is a decomposition into
eigenspace $\mathcal{E}=\oplus_{\lambda}\mathcal{E}[\lambda]$ where
$\lambda:T\to\mathbb{C}^{*}$ is a character and we have
$H^{i}_{T}(X,\mathcal{E})=H^{i}(X,\mathcal{E}[0])$.

###### Proof.

If $U\subset X$ is affine, then $\mathcal{E}_{\mid U}\cong E\times U$ for some
vector space $E$. The action of $T$ is given by an action on the first
component, and we get a decomposition $E\cong\oplus_{\lambda}E[\lambda]$ into
$T$-eigenspaces. Hence $\mathcal{E}_{\mid
U}\cong\oplus_{\lambda}\mathcal{E}_{\mid U}[\lambda]$, and we get a global
decomposition $\mathcal{E}\cong\oplus_{\lambda}\mathcal{E}[\lambda]$ since the
$T$ -action does not depend on $U$. Moreover, there is an isomorphism of
functors $\Gamma_{T}(X,\mathcal{E})\cong\Gamma(X,\mathcal{E}[0])$. It follows
that $H^{i}_{T}(X,\mathcal{E})=H^{i}(X,\mathcal{E}[0])$ by taking derived
functors. ∎

Most of these results were used implicitly in [1].

###  2.2 Kontsevitch’s theorem

Let $X$ be a smooth algebraic variety over the complex numbers. The Hochschild
cohomology of $X$ is defined as
$HH^{\bullet}(X)=\bigoplus_{i}\text{Ext}^{i}_{\mathcal{O}_{X\times
X}}(\mathcal{O}_{\Delta},\mathcal{O}_{\Delta})$ where $\Delta\subset X\times
X$ is the diagonal. It contains information about deformations of $X$, for
example $H^{1}(X,TX)$ is the space parametrizing complex deformations of $X$.
Let us recall two important theorems about the Hochschild cohomology of a
smooth complex algebraic variety. The first theorem is the _Hochschild-
Kostant-Rosenberg isomorphism_

######  Theorem 2.14.

([7],) Let $\iota:\Delta\to X\times X$ be the diagonal inclusion. There is a
quasi-isomorphism

$\iota^{*}\mathcal{O}_{\Delta}\cong\bigoplus_{i\in\mathbb{N}}\Omega_{X}^{i}[i]$

in the derived category $D^{b}(\text{Coh}(X))$.

Here $\Omega_{X}^{i}$ is the sheaf of differential forms on $X$ of degree $i$,
and $[i]$ is the cohomological shift functor.

This is a stronger version of the HKR theorem (that appeared first in [9])
stating that there is an isomorphism of graded vector spaces

$I_{HKR}:HH^{\bullet}(X)\to
HT^{\bullet}(X):=\bigoplus_{i+j=\bullet}H^{i}(X,\wedge^{j}TX))$

Let us recall the proof of theorem 2.14 : a non-trivial argument using sheaves
(see [17], [7]) reduces it to the statement when $X=\text{Spec}(R)$ is affine.
This is the case treated in [9]. We use the _bar complex_
$\mathscr{C}_{\bullet}(R)$, which is a flat resolution of
$\mathcal{O}_{\Delta}$ by $\mathcal{O}_{X\times X}$-modules. Recall that the
bar resolution is given by $\mathscr{B}_{i}(R):=R^{\otimes(i+2)}$. We define
$\mathscr{C}_{i}(R)=R\otimes_{R\times R}\mathscr{B}_{i}(R)$. The differential
on $\mathscr{B}_{\bullet}(R)$ is given by

$d(1\otimes a_{1}\otimes\dots\otimes a_{i}\otimes 1)=a_{1}\otimes
a_{2}\dots\otimes a_{i}\otimes 1-1\otimes a_{1}a_{2}\otimes
a_{3}\otimes\dots\otimes a_{i}\otimes 1+\dots+(-1)^{i+1}1\otimes
a_{0}\otimes\dots\otimes a_{i}$

A quasi-isomorphism
$I:\mathscr{C}_{\bullet}(R)\to(\bigwedge^{\bullet}(\Omega_{R}^{1}[1]),0)$ is
given by

$1\otimes(1\otimes a_{1}\otimes\dots\otimes a_{i}\otimes 1)\mapsto
da_{1}\wedge da_{2}\dots\wedge da_{i}$

Since the bar complex resolves $R$ as $R$-bimodule, this bar complex is
isomorphic to $\mathcal{O}_{\Delta}$ in $D^{b}(X\times X)$. This finishes the
proof.

In general, $I_{HKR}$ is not multiplicative. To understand how the
multiplications are related, we need to introduce the Todd class. We use the
formalism of Atiyah class explained in section $1$ of [10].

There is an exact sequence

$0\to\Omega_{X}\otimes TX\to J^{1}(TX)\to TX\to 0$

where $J^{1}(TX)$ is the bundle of first-order jet of $TX$. Such sequence
gives a class in $Ext^{1}(TX,\Omega_{X}\otimes TX)=H^{1}(X,\Omega_{X}\otimes
End(TX))$.

######  Definition 2.15.

The _Atiyah class_ of $X$ is defined as the class of this extension :
$At(X)\in H^{1}(X,\Omega_{X}\otimes\text{End}(TX))$.

######  Definition 2.16.

The _Todd class_ of $X$ is defined as

$Td(X)=\det\left(\frac{At(X)}{1-e^{-At(X)}}\right)$

A more familiar definition of the Todd class involves Chern classes of $TX$.
Our definition is equivalent to that, according to the following statement
(see [10], formula $1.4.1$) :

######  Proposition 2.17.

We have $tr(\wedge^{i}At(X))=c_{i}(X)$ where $c_{i}(X)$ is the $i$-th Chern
class of the tangent bundle $TX$.

Now we can state Kontsevitch’s theorem :

######  Theorem 2.18.

[11][4] Let $X$ be as before and $t$ be the Todd class of $X$. Then, the
composition

$\langle t^{-1/2},-\rangle\circ I_{HKR}:HH^{\bullet}(X)\to HT^{\bullet}(X)$

is a ring isomorphism.

###  2.3 Equivariance of the twisted HKR isomorphism

We now assume that $T$ is a complex torus acting on a smooth variety $X$.

######  Lemma 2.19.

If $X$ is affine, the isomorphism $I_{HKR}$ is $T$-equivariant.

###### Proof.

Say $X=\text{Spec}(R)$. Let us keep notation from the discussion after 2.14.
The differential of the bar-complex is $T$-equivariant, hence the bar-complex
gives a flat $T$-equivariant resolution of $\mathcal{O}_{\Delta}$. Moreover
the map $I:A\to\Omega_{R},a\mapsto da$ is $T$-equivariant, where $\Omega_{R}$
is the module of Kähler differential. This is because there is a natural
identification $\Omega_{R}\cong J/J^{2},da\mapsto a\otimes 1-1\otimes a$,
where $J$ is the kernel of the multiplication map $R\otimes R\to R$. Under
this identification, the map $a\mapsto da$ is $T$-equivariant. It follows that
the quasi-isomorphism
$I:\mathscr{C}_{\bullet}(R)\to(\bigwedge^{\bullet}(\Omega_{R}^{1}[1]),0)$ is
also $T$-equivariant. ∎

######  Lemma 2.20.

If $X$ is a quasi-projective smooth variety acted upon by $T$ then the
previous lemma holds.

###### Proof.

By Sumihiro’s theorem, there is an $T$-invariant affine open cover. So we can
check the statement on each affine open set $U\subset X$ that is
$T$-invariant. But recall that in the discussion sketching the proof of
theorem 2.14, we saw that we could glue together the bar complex to construct
a global quasi-isomorphism $I_{HKR}$. So it means that the invariance can be
only checked for the bar complex $\mathscr{C}_{\bullet}(R)$, that was
precisely given by our previous lemma. Hence $I_{HKR}$ is $T$-equivariant. ∎

######  Lemma 2.21.

If $X$ is as before, then the Atiyah class of $X$ is $T$-invariant. Therefore
the Todd class of $X$ is also $T$-invariant.

###### Proof.

This follows from lemma 2.12 because the exact sequence defining the Atiyah
class is $T$-equivariant. ∎

It follows that the twisted HKR morphism $HH^{\bullet}(X)\to HT^{\bullet}(X)$
is $T$-equivariant, which completes the proof of our theorem 1.2. It would be
interesting to generalize theorem 1.2 and relate it with the geometry of
$X/G$, where $G$ is a complex reductive group. However it is known that even
when $G$ is a finite group, the analogue of Kontsevitch’s theorem does not
hold for the stack $[X/G]$, see [15]. Without Sumihiro’s theorem,(i.e without
the existence of an invariant cover) it seems non-trivial to prove that
$I_{HKR}$ is $G$-equivariant.

##  3 Applications to the small quantum group

We recall notations and results from [1] and prove our main result, theorem
1.5. As an application, we show that the multiplication on the subalgebra
generated by the Harish-Chandra center and the Poisson bi-vector field $\tau$
is untwisted.

###  3.1 The small quantum group

We summarize results from [1]. Let $G$ be a semisimple Lie group of adjoint
type, $H\subset B$ a Cartan subgroup contained in a Borel subgroup and $N$ the
unipotent radical of $B$. We write $\mathfrak{g},\mathfrak{b},\mathfrak{n}$
for the corresponding Lie algebras. If $q$ is a primitive $\ell$-th root of
unity where $\ell$ is odd, greater that the Coxeter number of $\mathfrak{g}$,
prime to the index of connection, and prime to $3$ if there is a component of
type $G_{2}$. To this data Lusztig associated a canonical finite-dimensional
Hopf algebra $\mathfrak{u}_{q}(\mathfrak{g})$, called the _small quantum
group_ , see e.g [14]. An important problem in representation theory is to
understand its center. Since $\mathfrak{u}_{q}(\mathfrak{g})$ splits as sum of
two-sided ideals, _blocks_. They are parametrized by the orbit of a certain
extension of the Weyl group in the weight lattice. So it is enough to
understand structure of the center for each block. The regular block that
contains the trivial representation is the _principal block_ (corresponding to
$\lambda=0$), denoted by $\mathfrak{u}_{0}(\mathfrak{g})$ or
$\mathfrak{u}_{0}$ for simplicity.

A major advance was obtained in [1] where the center was described
geometrically via the Springer resolution. Before stating the main result, let
us introduce some notations : $z$ is the trivial line bundle on
$\widetilde{\mathcal{N}}$ with the $\mathbb{C}^{*}$-action given by
$(z,v)\mapsto z^{2}v$, $R$ is the principal block, viewed as bimodule over
$\mathfrak{u}$. By definition
$H^{i}(\widetilde{\mathcal{N}},(\wedge^{j}T\widetilde{\mathcal{N}})^{k}):=H^{i}(G/B,(pr_{*}\wedge^{j}T\widetilde{\mathcal{N}})^{k})$.
By definition $HH^{\bullet}(\mathfrak{u}_{0})$ is the graded ring

$\bigoplus_{m}\mathrm{Hom}_{D^{b}(\mathfrak{u}_{0}\otimes\mathfrak{u}_{0}^{op})}(R,R[m])$

where the multiplication is given by the composition. Moreover,
$HH^{*}_{\mathbb{C}^{*}}(\widetilde{\mathcal{N}})$ is by definition the ring

$\bigoplus_{j+k=m}\mathrm{Hom}_{D^{b}(\mathrm{Coh}^{\mathbb{C}^{*}}(\widetilde{\mathcal{N}}\times\widetilde{\mathcal{N}}))}(\mathcal{O}_{\Delta},\mathcal{O}_{\Delta}[j]\otimes
z^{k})$

where the ring structure comes from the Yoneda product. The main theorem in
[1] can be formulated :

######  Theorem 3.1.

There is a ring isomorphism $HH^{\bullet}(\mathfrak{u}_{0})\cong
HH^{*}_{\mathbb{C}^{*}}(\widetilde{\mathcal{N}})$.

We can now give a proof of the theorem 1.5 :

######  Theorem 3.2.

The composition

$HH^{\bullet}(\mathfrak{u}_{0}(\mathfrak{g}))\overset{BL}{\to}\bigoplus_{i+j+k=\bullet}H^{i}(\widetilde{\mathcal{N}},\wedge^{j}T\widetilde{\mathcal{N}})^{k}\overset{\langle-,Todd(\widetilde{\mathcal{N}})^{-1/2}\rangle}{\to}\bigoplus_{i+j+k=\bullet}H^{i}(\widetilde{\mathcal{N}},\wedge^{j}T\widetilde{\mathcal{N}})^{k}$

is a ring isomorphism.

###### Proof.

Let us recall how to deduce the isomorphism 1.1 from the theorem 3.1. There
are isomorphisms

$HH^{\bullet}(\mathfrak{u}_{0})\overset{\ref{thmBL}}{\cong}\bigoplus_{q+k=\bullet}Ext^{q}_{\mathrm{Coh}^{\mathbb{C}^{*}}(\widetilde{\mathcal{N}}\times\widetilde{\mathcal{N}})}(\mathcal{O}_{\Delta},\mathcal{O}_{\Delta}\otimes
z^{k})\cong\bigoplus_{i+j+k=\bullet}H^{i}_{\mathbb{C}^{*}}(\widetilde{\mathcal{N}},\wedge^{j}T\widetilde{\mathcal{N}}\otimes
z^{k})$

Here
$H^{i}_{\mathbb{C}^{*}}(-)=R^{i}(Hom_{\mathbb{C}^{*}}(\mathcal{O}_{X},-))$.
The second isomorphism follows by taking the $\mathbb{C}^{*}$ invariant on
both side of the usual HKR isomorphism (and using corollary 2.10). Hence we
know that

$H^{i}_{\mathbb{C}^{*}}(\widetilde{\mathcal{N}},\wedge^{j}T\widetilde{\mathcal{N}}\otimes
z^{k})=(H^{i}(\widetilde{\mathcal{N}},\wedge^{j}T\widetilde{\mathcal{N}}\otimes
z^{k}))^{\mathbb{C}^{*}}=(H^{i}(G/B,pr_{*}\wedge^{j}T\widetilde{\mathcal{N}}\otimes
z^{k}))^{\mathbb{C}^{*}}$

since $pr:\widetilde{\mathcal{N}}\to G/B$ is an affine
$\mathbb{C}^{*}$-morphism. Since the $\mathbb{C}^{*}$-action is trivial on
$G/B$ by proposition 2.13 we obtain an isomorphism

$(H^{i}(G/B,pr_{*}\wedge^{j}T\widetilde{\mathcal{N}}\otimes
z^{k}))^{\mathbb{C}^{*}}\cong
H^{i}(G/B,(pr_{*}\wedge^{j}T\widetilde{\mathcal{N}}\otimes z^{k})[0])\cong
H^{i}(G/B,(pr_{*}\wedge^{j}T\widetilde{\mathcal{N}})^{-k})$

Putting everything together we obtain indeed the isomorphism stated in Theorem
1.1 :

$HH^{\bullet}(u_{0})\cong\bigoplus_{q+k=\bullet}Ext^{q}(\mathcal{O}_{\Delta},\mathcal{O}_{\Delta}\otimes
z^{k})\cong\bigoplus_{i+j+k=\bullet}H^{i}(\widetilde{\mathcal{N}},(\wedge^{j}T\widetilde{\mathcal{N}})^{k})$

Now the isomorphism
$HH^{\bullet}(\mathfrak{u}_{0})\cong\bigoplus_{q+k=\bullet}Ext^{q}(\mathcal{O}_{\Delta},\mathcal{O}_{\Delta}\otimes
z^{k})$ is multiplicative. Since the Todd class is $T$-invariant by lemma
2.21, and that $I_{HKR}$ is $T$-equivariant by lemma 2.20 it follows that the
usual multiplicative HKR theorem restricts to a multiplicative isomorphism

$\bigoplus_{q+k=\bullet}Ext^{q}(\mathcal{O}_{\Delta},\mathcal{O}_{\Delta}\otimes
z^{k})\cong\bigoplus_{i+j+k=\bullet}H^{i}_{\mathbb{C}^{*}}(\widetilde{\mathcal{N}},\wedge^{j}T\widetilde{\mathcal{N}}\otimes
z^{k})$

Finally the isomorphism

$\bigoplus_{i+j+k=\bullet}H^{i}_{\mathbb{C}^{*}}(\widetilde{\mathcal{N}},\wedge^{j}T\widetilde{\mathcal{N}}\otimes
z^{k})\cong\bigoplus_{i+j+k=\bullet}H^{i}(G/B,(pr_{*}\wedge^{j}T\widetilde{\mathcal{N}})^{k})$

is multiplicative, because for an affine map $q:X\to Y$, the equivalence
$\text{Coh}(X)\cong\text{Coh}(Y)-q_{*}\mathcal{O}_{X}$ is monoidal, hence
respect the cup-product. ∎

###  3.2 Multiplicative structure of $HC[\tau]$

We will compute the multiplicative structure of a subalgebra of
$z_{0}(\mathfrak{g})$, already noticed in [12], where the arguments was given
at level of sheaf cohomology.

######  Lemma 3.3.

If $Y=T^{*}X$ is a cotangent bundle of a smooth algebraic variety $X$, with
projection morphism $pr:Y\to X$, then the Todd class of $Y$ is given by the
pullback to $Y$ of

$\prod_{\alpha,\beta\in
C}\frac{-\alpha\beta}{(1-e^{-\alpha})(1-e^{-\beta})}\in H\Omega^{\bullet}(X)$

where $C$ is the set of Chern roots of $TY$. In particular, this Todd class
can be written $1+pr^{*}a$ where $a\in H^{\geq 2}(X)$.

###### Proof.

The map $pr:Y\to X$ induces a short exact sequence

$0\to T_{Y/X}\to TY\to pr^{*}TX\to 0$

Moreover, $T_{Y/X}\cong pr^{*}(T^{*}X)$. Using the naturality of the Todd
class and the additivity on short exact sequences the result easily follows. ∎

For example if $X=G/B$ then $C$ is the set of positive roots of $\mathfrak{g}$
and the Todd class of $G/B$ is given by

$Todd(G/B)=\prod_{\alpha\in\Phi^{+}}\frac{\alpha}{1-e^{-\alpha}}$

Let $\tau$ be the the generator of
$H^{0}(\widetilde{\mathcal{N}},End(T\widetilde{\mathcal{N}}))$ given by the
identity map.

######  Definition 3.4.

We define $HC[\tau]$ to be the subalgebra of $z_{0}(\mathfrak{g})$ generated
by the Harish-Chandra center $HC_{\lambda}$ and $\tau$.

Of course, for singular blocks $\mathfrak{u}_{\lambda}(\mathfrak{g})$ there is
a similar subalgebra inside $z_{\lambda}(\mathfrak{g})$, also written
$HC[\tau]$ if there is no confusion.

######  Proposition 3.5.

For any block, the multiplicative structure of $HC[\tau]$ is untwisted.

###### Proof.

Let us take $U\subset G/P$ an affine open set isomorphic to an affine space.
If $pr:\widetilde{\mathcal{N}}_{P}\to G/P$ is the projection, then
$pr^{-1}(U)$ is an affine space with coordinates $(x,y)$. In these coordinates
the Poisson bivector field is $\tau=\sum_{i}\frac{\partial}{\partial
x_{i}}\wedge\frac{\partial}{\partial y_{i}}$ and the Todd class is $t=1+\sum
a_{ij}dx_{i}\wedge dx_{j}+\text{higher order terms}$. It follows that
$(1-t,\tau)=0$ where $(\cdot,\cdot)$ is the pairing
$T^{*}\widetilde{\mathcal{N}}_{P}\otimes
T\widetilde{\mathcal{N}}_{P}\to\mathcal{O}_{\widetilde{\mathcal{N}}}$. In the
exact sequence

$0\to pr^{*}T^{*}(G/P)\to\widetilde{\mathcal{N}}_{P}\to pr^{*}T(G/P)\to 0$

the left hand-side corresponds to the fiber of $pr$ since
$\widetilde{\mathcal{N}}_{P}=T^{*}(G/P)$. In particular an element $\alpha\in
HT^{\bullet}(\widetilde{\mathcal{N}})$ coming from
$H^{\bullet}(X,\Omega^{\bullet})$ can we written in local coordinates as a
polynomial in $\frac{\partial}{\partial y_{i}}$. For such a class it is again
clear that $(1-t,\alpha)=0$. In particular it holds for the Harish-Chandra
center $HC\cong H^{\bullet}(G/P)$. ∎

######  Corollary 3.6.

If $G/P$ is a projective space $\mathbb{P}^{n}$, then
$HH^{0}(\mathfrak{u}_{\lambda}(\mathfrak{g}))=z_{\lambda}(\mathfrak{g})$ is
the polynomial algebra $\mathbb{C}[x,\tau]/(x^{a}\tau^{b}:a+b>n)$.

###### Proof.

It follows immediately from the previous proposition and the fact (proved in
[13]) that $HC[\tau]=z_{\lambda}(\mathfrak{g})$ if $G/P_{\lambda}$ is a
projective space. ∎

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