A geometric realization of the asymptotic affine Hecke algebra
Abstract.
A key tool for the study of an affine Hecke algebra is provided by Springer theory of the Langlands dual group via the realization of as equivariant -theory of the Steinberg variety. We prove a similar geometric description for Lusztig’s asymptotic affine Hecke algebra identifying it with the sum of equivariant -groups of the squares of -fixed points in the Springer fibers, as conjectured by Qiu and Xi (the same result was also obtained by Oron Popp using different methods). As an application, we give a new geometric proof of Lusztig’s parametrization of irreducible representations of . We also reprove Braverman-Kazhdan’s spectral description of . As another application, we prove a description of the cocenters of and conjectured by the first author with Braverman, Kazhdan and Varshavsky. The proof is based on a new algebraic description of , which may be of independent interest.
1. Introduction
1.1. Affine Hecke algebra and asymptotic affine Hecke algebra
Let be a reductive algebraic group over . Let be the Weyl group of and let be the coweight lattice of . The (extended) affine Weyl group of is defined as . Let be the affine Hecke algebra of , recall that is a deformation of the group algebra of over , where and are formal variables. Let be the complexification of . For we will denote by the corresponding specialization of .
Kazhdan-Lusztig and Ginzburg showed that has a geometric realization: it can be identified with the convolution algebra , where is the Springer resolution for the Langlands dual group . Kazhdan and Lusztig used the relation between and geometry of the Steinberg variety to parametrize simple -modules for not a root of unity. It turns out that this parametrization extends to any that is not a root of the Poincaré polynomial of (see [61]), moreover, the parametrization does not depend on .
Lusztig introduced the (based) algebra called the asymptotic Hecke algebra, its basis is parametrized by , and there is an injective homomorphism . We set .
Informally, one can think about the algebra as the ‘‘limit’’ of as goes to . As discovered by Lusztig, representations of are closely related to those of , more precisely pulling back an irreducible -module under the homomorphism , one gets a so-called standard module over 111see [39], [40] for the case of generic ; more generally, from [61, Theorem 3.2] and [19, Corollary 2.6] it follows that this holds for such that . We reprove and strengthen this fact below. This connection explains the observation above that the parametrization of modules over the Hecke algebra remains independent of the parameter , as long as is not a root of .
The goal of this paper, achieved in Theorem A, is to prove a similar geometric description of the algebra conjectured in [51] and derive some applications (Theorems B, C etc.). We note that Theorem A was also proved by Oron Popp in his PhD thesis [50] by a different method.
The coherent realization of is a manifestation of local Langlands duality used in the proof of a special case of local Langlands conjectures [36]. It would be very interesting to find an interpretation and a generalization for Theorem A in that context. In particular, according to a recent insight of Braverman and Kazhdan [19], can be viewed as a ring of Iwahori bi-invariant distributions on the -adic group intermediate between the algebra of compactly supported distributions (well-known to be isomorphic to a specialization of ) and Harish-Chandra Schwartz space of tempered distributions. In [19] one also finds a generalization of this definition to not necessarily Iwahori bi-invariant distribution. We hope that Theorem A admits an (at least conjectural) extension to that generality.
1.2. Filtrations on and two-sided cells
Let us now return to the identification . The group acts on with a finite number of orbits ordered by the closure order. For every orbit consider its closure and take its preimage in . The -equivariant -theory of this preimage is a term in the filtration of by the two-sided ideals indexed by nilpotent orbits . Each subquotient is a bimodule over that is clearly isomorphic to , where is the fiber of the Springer resolution over called the Springer fiber, is the reductive part of the centralizer of in and the -action on is defined using the Jacobson-Morozov Theorem. Set , it acts naturally on .
Remark 1.2.1.
Note that is also a ring (without a unit element). To see the ring structure geometrically, we need to identify with , where is the Slodowy variety corresponding to and is the -group of the category of -equivariant coherent sheaves on supported on , see [56, Theorem B.2].
There is a way to describe the filtration above algebraically. Let be the canonical basis of introduced in [35] and generalized by Lusztig to the case of extended Weyl groups in [39]. Let be the minimal based (i.e., spanned over by a subset of the canonical basis) two-sided ideal of that contains . In [37] Lusztig introduced a notion of the two-sided cell in that can be characterized as follows: two elements lie in the same two-sided cell iff . We obtain a partition of into two-sided cells. To a cell there thus corresponds the two-sided ideal . This is the so-called cell filtration. Let us denote by the corresponding subquotient.
We obtain a partial order on the set of two-sided cells defined as follows: iff . By a result of Lusztig (see [40]), the set of -orbits in is in bijection with the set of two-sided cells in . Let us denote this bijection by . It then follows from results of Xi [62] and of the first author [10, Theorem 55, §11.3] that if then and the bijection above is order-preserving. Thus the cell filtration will also be referred to as the geometric filtration.
1.2.1. The direct sum decomposition for the ring
The ring can be decomposed as the direct sum , where runs over the set of two-sided cells in .
For we have a natural homomorphism of algebras . This homomorphism becomes an isomorphism after tensoring by . Let be the composition of and the projection onto .
1.3. Main results
1.3.1. Geometric description of
One of the main results of this paper is a geometric description of conjectured by Qiu and Xi in [51], see also Propp’s thesis [50, Theorem 1.5.2] for another proof.
Theorem A.
There exists an isomorphism of rings .
Remark 1.3.1.
Note that does not depend on , this corresponds to the fact that acts trivially on . Specializing at we obtain an identification ; however, specializing at yields another isomorphism.
Remark 1.3.2.
Let us briefly describe the idea of the proof of Theorem A. Recall the (injective) homomorphisms , . They induce -bimodule and -bimodule structures on . Moreover, considered as a left (resp. )-module, it is isomorphic to (see Corollary 2.3.2). We then prove the following theorem.
Theorem B.
The right action of induces isomorphisms of algebras
where stands for the set of endomorphisms, which are both -and -linear.
Warning 1.3.3.
Note that is not isomorphic to because is a ring without a unit. We only have the natural embedding induced by the right multiplication.
Thus, in order to establish the identification between and we need to construct a homomorphism and prove that it is an isomorphism. To construct the aforementioned homomorphism, we need to find a -bimodule that is isomorphic to as a -module. The natural candidate is . We prove that is indeed isomorphic to . To see that, we use the Bialynicki-Birula type decomposition of by the attractors via the -action studied by De Concini, Lusztig and Procesi in [26]. In more detail, recall that if is a smooth projective variety with a -action then by the Bialynicki-Birula theorem attractors to the connected components of are affine fibrations. It follows that the decomposition of by the attractors induces a filtration on the (complexified) -theory of with associated graded being isomorphic to the -theory of if is generated by algebraic cycles. Variety is projective but not smooth, so the Bialynicki-Birula theorem can not be applied to it directly. On the other hand by the results of [26], is smooth and attractors in of the components of are indeed affine fibrations. So, we obtain a filtration on with associated graded being isomorphic to . We then produce a splitting of this filtration. To this end we consider closures of the subvarieties , where are connected components of . We then consider natural morphisms , and the splitting is given by . In other words, the splitting is given by the natural correspondences .
So, we obtain a homomorphism , and it remains to check that it is an isomorphism. Injectivity is easy since we have an identification (of right -modules), and our homomorphism is then given by the action of on itself via right multiplication. To check surjectivity, it is enough to show that any element of is uniquely determined by its value on . To see that we use the identifications
and the only thing to check is that the image of in after these identifications is a (left) invertible element (we use Theorem B). It is clearly right invertible, but is left-Noetherian, so right invertible elements are left invertible.
1.3.2. Representation theory of from the geometric perspective
Realization of as the convolution algebra allows one to apply geometric methods to representation theory of (cf. [36] and [20, Sections 7, 8]). Our geometric realization has particularly favorable properties since is smooth and projective, and its homology is generated by algebraic cycles ([26]).
In [39] Lusztig classified irreducible modules over the algebra . He proved that these modules are parametrized by the triples (up to conjugation), where is a nilpotent element, is a semisimple element of and is an irreducible representation of . Irreducible -module corresponding to will be denoted by . For a variety equipped with the action of an algebraic group we set . We reprove Lusztig’s description using our geometric approach by proving the following theorem.
Theorem C.
(a) Irreducible modules over are all of the form .
(b) We have .
Let us briefly outline the argument. Part of the Theorem follows from general considerations about modules over convolution algebras (see Section A.2 for the details). To prove part , we recall that Lusztig’s parametrization works as follows (see [40]): is the unique irreducible module over such that is isomorphic to for a generic . So, in order to prove part we need to check that for a generic . We prove that this holds for any . This implies part of the Theorem and also shows that for every (this result is new for being a root of ).
1.3.3. Proof of Braverman-Kazhdan’s theorem
In [19] the authors described the algebra in spectral terms by formulating a version of the matrix Paley-Wiener theorem for (see [19, Theorem 1.8 (3)]). We reprove their theorem using our geometric approach.222We were not able to follow some steps in the argument of [19] (see footnotes , below), we fill in the details not found in loc. cit using our present geometric methods. Stefan Dawydiak [24] developed an alternative algebraic approach to completing the proof of that theorem. Let us first recall the content of [19, Theorem 1.8 (3)].
Pick a semisimple element . Let be a Levi containing and such that . In [19, Section 1.2] the authors consider a certain family of -modules over that we will denote by . Its fiber over is .
Theorem 1.3.4 (Braverman-Kazhdan).
Let be a subalgebra of (where the product is taken over all compact ) given by the following conditions:
a) any does not have poles at the points of families which correspond to (cf. loc. cit.) non-strictly positive characters of Levi subgroups;
b) the endomorphisms are compatible in some precise sense (see Section 6.2 for details).
Then .
Before sketching our proof of this theorem, let us discuss certain families of modules over , , that should be considered as geometric counterparts of the Braverman-Kazhdan’s families .
Here by a ‘‘geometric’’ family we mean the result of the following construction. Let be a torus and let be the corresponding Levi. Let be an irreducible representation of . Set (the smallest closed diagonalizable subgroup of containing and ). We have two families over :
The algebras , act naturally on and the algebra acts naturally on . We also have the natural action of on the families , . Taking -multiplicity spaces we obtain the families , . One can show that for we have -equivariant identifications:
so these families are nothing but the families of (directs sums of) standard or irreducible modules over our algebras. We prove the following theorem that should be considered as a version ‘‘in families’’ of the identification above333Existence of an action of on that is algebraic in is claimed and used in [19], see the proof of Theorem 2.4 in loc. cit., but not checked there in detail. Our Theorem D fills in the details.
Theorem D.
We have a natural -equivariant isomorphism of the families of and -modules respectively:
Let us now return to the proof of Theorem 1.3.4. We show that the families , can be identified after restricting to some open dense subset containing all non-strictly positive characters. Using Theorem D, it then follows that we are reduced to proving the following proposition.
Proposition 1.3.5.
Let be the subalgebra of consisting of elements satisfying the following property:
for any conjugate pair (here for a Levi subgroup , so that , and ), the following equality holds:
Then via the action map .
To prove this proposition, we show that the homomorphism becomes an isomorphism after completion at every point of by describing completions of , as explicit subalgebras of the completion of .444Our argument is largely parallel to the original proof of [19], the difference is that we consider completion at the points of which is an exact functor, so, in particular, the natural embedding remains embedding after completions. In general it does not induce an embedding of fibers (since the fibers of are not semisimple in general, see Section 4.4.2), so the last paragraph of the proof of [19, Theorem 1.8] does not hold as stated. We use our geometric description of and localization theorem in -theory to analyze the completion of .
1.3.4. Description of the cocenter of
In the last part of the text we use our geometric approach to to obtain a description of the cocenter of conjectured in [12, Section 6.2]. Let us formulate the theorem.
Let be the commuting variety for (with the reduced scheme structure). Set .
Theorem E.
Let be the space consisting of regular functions on , subject to the following properties:
a) for any semisimple , is locally constant (and, hence, gives a well-defined function on the component group of ;
b) is a sum of characters of the group arising in .
Then .
1.4. Structure of the paper
The paper is organized as follows. In Section 2 we recall definitions and known properties of the algebras , . Section 3 is devoted to the proof of Theorem B. In Section 4 we prove Theorem A. In Section 5 we study representations of , construct families of (irreducible) modules over it and prove Theorems C, D. In Section 6, we prove [19, Theorem 1.8 (3)] by our methods. In Section 7 we prove Theorem E i.e. describe the cocenter of . Appendix A contains proofs of various general facts about equivariant -theory that we need for our arguments.
1.5. Acknowledgements
We gratefully acknowledge helpful input from Dan Ciubotaru, Stefan Dawydiak, Pavel Etingof, Michael Finkelberg, Victor Ginzburg, Mikhail Goltsblat, William Graham, Do Kien Hoang, David Kazhdan, Ivan Losev, Jakub Löwit, Victor Ostrik, Oron Propp, Vadim Vologodsky and Zhiwei Yun. We are grateful to George Lusztig for useful comments on the first version of the text.
I. K. especially thanks Dmitrii Zakharov for explaining the material of Proposition 6.6.1 and Michael Finkelberg for his TeX-nical generosity.
R.B. was partly supported by the NSF grant DMS-2101507.
2. Generalities on and
2.1. The affine Hecke algebra
Let be as above and let be the set of simple reflections. Let be the length function on . The algebra is an algebra over with basis . Multiplication in this algebra is determined by the relations when and for . Recall that is the complexification of .
2.1.1. Lusztig’s cells
There is a partition of into the union of the so-called cells. We start with a brief recollection of it.
Let be the strong Bruhat order. First of all, two elements and of are said to be connected, if either or ; and : in particular, is odd. (Here, we denote by the corresponding Kazhdan-Lusztig polynomial, see [35] and [39, Section 1.2].)
To each corresponds the so-called left descending set .
Now, for we say that , if there is a chain of elements , s.t. all neighbours in it are connected, and .
It is well-known that is a partial preorder, and the corresponding equivalence classes are called left cells: this is the work by Lusztig ([37]).
Right cells are subsets of the form for a left cell. Their definition can also be given in a similar fashion to the one above. We should only replace by .
Finally, there are also two-sided cells (we will denote the set of them by ). They are defined in almost the same fashion as right or left ones; now the appropriate condition for neighbors in is that either or for each .
2.1.2. Two-sided cells and nilpotent conjugacy classes
The famous theorem by Lusztig (cf. [40]) says that two-sided cells in are in bijection with the conjugacy classes of nilpotent elements in the Lie algebra of the Langlands dual group . In particular, this gives an order on cells (via the closure order on the nilpotent orbits). This order coincides with the natural one (this was conjectured by Lusztig and proved by the first author in [9, Theorem 4 (b)]).
The so-called -function sends every two-sided cell to the dimension of the corresponding Springer fiber .
It has a combinatorial meaning as well. Namely, recall that is the affine Hecke algebra of and let us denote by the Kazhdan-Lusztig basis elements in . Then, if are structure constants of with respect to , is the lowest possible degree of non-zero term in , , .
2.2. Asymptotic affine Hecke algebra
Now we recall Lusztig’s definition of the asymptotic affine Hecke algebra (cf. loc. cit.). It has a basis . In this basis the structure constant is the constant term of the polynomial : .
It is shown in loc. cit. that is actually an associative algebra. Moreover, denoting the span over of by , one gets a decomposition of as a direct sum (product) of rings:
We will sometimes denote by for corresponding to the nilpotent orbit (cf. Section 2.1.2). The same notation will be used for subscripts and superscripts of various morphisms in Section 2.3.
2.2.1. Distinguished involutions and the unit element in
The set of distinguished involutions: consists of all elements , such that It is known that its elements are actually involutions, and each one-sided cell contains exactly one of them.
The element is the unit element in the algebra , while for , the element is the unit element in .
2.3. Relation between and
For a two-sided cell we set
Obviously is the corresponding term of the cell filtration.
The coincidence of the geometric and the cell filtration implies that can be identified with the -equivariant -theory of the preimage of in and , see [56], [62] for details.
Finally, note that homomorphism induces homomorphisms
( is the composition of and the projection to the direct summand).
The isomorphism restricts to the isomorphisms .
2.3.1. Bimodule structure on
A homomorphism of rings equips with the structure of an bimodule. In particular, the homomorphism defines a - bimodule structure on . Similarly, defines a - bimodule structure on .
In this section, we will give an alternative description of the (left) action of on as above. We will then conclude that this -module is isomorphic to . We will obtain similar statements for the -module .
Let denote the projection . Following [41, Section 3] let us define the action of on by the formula:
| (2.3.1) |
Note that directly from the definitions is the isomorphism of -modules (in particular, it follows that the formula (2.3.1) indeed defines the action of ).
Note now that is an -submodule of . Modding out by this submodule, we obtain the action of on such that for we have:
| (2.3.2) |
It induces the action .
Lemma 2.3.1.
(a) Two -actions on defined above (the one coming from and the one given by the formula (2.3.2)) coincide.
(b) Two -actions on defined above (the one coming from and the one given by the formula (2.3.2)) coincide.
Proof.
It is enough to prove part (a), part (b) will follow.
Let us prove part (a). We need to check that
for and . So, our goal is to check that
| (2.3.3) |
Note that directly from the definitions we have:
| (2.3.4) |
The following formula follows from [38, 2.4 (d)] (see also [12, Proof of Proposition 2]):
| (2.3.5) |
Now, setting , , summing (2.3.5) over and using (2.3.4) together with the fact that is the identity element of we obtain the desired equation (2.3.3). ∎
Corollary 2.3.2.
(a) The -module defined via is isomorphic to .
(b) The -module defined via is isomorphic to .
Proof.
It follows from Lemma 2.3.1 that is the desired isomorphism of (resp. ) - modules. ∎
2.4. Description of via finite centrally extended sets
A key ingredient in our proof of the Theorem B presented in the next section is the finite-set realization of , which we now recall.
The following was conjectured (in a slightly different form) by Lusztig [40]. It was proven in [60] in type A and in [15] in general.
Definition 2.4.1.
A centrally extended finite -set (for a reductive group ) is a finite -set equipped with some central extension for every such that for every the conjugation isomorphism extends to an isomorphism between these central extensions satisfying natural compatibilities.
For a centrally extended finite -set , one defines an equivariant sheaf on as a usual -equivariant sheaf with an additional data of -action on , which induces an action of -factor by identity character. This allows us to consider -algebra .
Theorem 2.4.2 ([15]).
Let be a two-sided cell in and let denote the corresponding nilpotent element. Then there exists a centrally extended finite -set such that .
In particular, one has a central embedding , which will be of crucial importance for us.
Remark 2.4.3.
For the situation simplifies. First, the centralizers in question are connected so the -action is trivial.
Also, it turns out that no non-trivial central extensions appear. Thus, we arrive at the following statement proved in [60]: can be realized as a matrix algebra over the ring . Note that corresponding to in our notations is corresponding to the extended affine Weyl group associated with in the notations of [60].
2.5. Description of and via bimodules over non-commutative Springer
We now recall, following [13], the relation between the above algebras and the non-commutative Springer resolution introduced in [7], [14].
Recall that is an algebra equipped with a derived equivalence
where the subscript denotes the equivariant structure. It has a canonical filtration indexed by the poset of nilpotent conjugacy classes in equipped with the closure order.
We will denote the associated graded piece by . It is known (cf. [14]) that is isomorphic to . Now, it follows from [13, Section 8] that there exists an isomorphism of -algebras:
| (2.5.1) |
where in the right hand side we take the -group of the category of semisimple bimodules. Here the morphism can be identified with the tautological isomorphism between the -theories of the categories of semisimple and all finite length modules. The -action on comes from the identification (2.5.1).
For future use we record the following.
Corollary 2.5.1.
The morphisms and are -linear.
Proof.
For this is immediate from the above. To deduce the claim for , recall that for any , , the sum running over distinguished involutions inside the cell associated to . Now the claim follows, since the -action commutes with both and the multiplication in . ∎
Remark 2.5.2.
See [39, Proposition 6] for a related statement.
Remark 2.5.3.
Note that is not a unital algebra, in particular, the -action on does not come from a homomorphism .
3. Proof of Theorem B
3.1.
Now we are ready to prove Theorem B. Recall that it claims that we have isomorphisms of algebras:
We have already observed in Corollary 2.3.2 that as left (and )-modules so it remains to check that the natural homomorphisms
| (3.1.1) |
are actually isomorphisms. Note that both of them are injective since is unital. So, it remains to check that the homomorpisms in (3.1.1) are surjective. We start from the first one, the proof for the second one is completely analogous.
First of all, we claim that it is enough to check that the homomorphism
| (3.1.2) |
is surjective. Indeed, since is a free -module, we have embeddings
Now the homomorphism (3.1.2) is given by , so if the operator lies in then .
So, we need to prove the surjectivity of (3.1.2). The proof consists of several steps.
A) Recall that is an algebra with a unit element (see Section 2.2.1). To prove the proposition it is sufficient to show vanishing of an -endomorphism of which sends to . This, in turn, can be reformulated as follows:
| (3.1.3) |
where abusing notations we denote the complexification of by the same letter.
B) To fix ideas, we first prove the equality (3.1.3) in the special case when . In this case centralizers of all nilpotent elements are connected, so has no zero-divisors. Let us denote by the image of the restriction homomorphism .
Now, since is injective and -linear (cf. [12], and Corollary 2.5.1), has the same (and, hence, ) rank as , thus it is an -submodule of full rank in . (These ranks are finite by [39, Proposition 6].)
Moreover, the bijective map is -linear as well (see Corollary 2.5.1), so .
It means that is -torsion.
Since has no zero divisors (centralizers being connected) and is a torsion-free module over (cf. Lemma 3.1.1 below) the statement follows
once we check that an -linear morphism is -linear.
The latter point follows from the well-known isomorphism between and the center of [5] and compatibility between the - and -actions on (cf. [60]).
C) We now consider the general case, then may have several connected components which we denote by , .
Each conjugacy class in gives rise to an idempotent in , so that , and (and, hence, ) has no zero-divisors. The desired statement follows as above from the following two lemmas.
Lemma 3.1.1.
is a torsion-free module over .
Lemma 3.1.2.
Any -linear (in particular, any -linear) endomorphism of preserves the decomposition
D) Proof of Lemma 3.1.1. In Section 2.4 we recalled the realization of as -equivariant -theory of a centrally extended finite set. We claim that the statement of Lemma 3.1.1 is true for any -module arising that way.
It suffices to consider the case when the action of on the finite set is transitive. If the corresponding central extension is trivial then the complexified equivariant -theory is identified with the ring of conjugation invariant regular functions on a finite index subgroup ; in general it is a subspace in the ring of invariant functions on a finite covering of such a subgroup . The statement is clear since a component in maps surjectively to a component in .
Now consider the -action on via (cf. Section 2.3.1), since ’s are orthogonal we get:
for . Thus consists of all elements in annihilated by for all , so the decomposition of from Lemma 3.1.2 is stable under -linear endomorphisms. ∎
This finishes the proof of Theorem B.
4. Proof of Theorem A
4.1. A homomorphism
4.1.1.
Set
Clearly, , is a (unital) algebra w.r.t. the natural convolution product. Our goal in this section is to provide an injective homomorphism ; it will be shown in Section 4.2 that is an isomorphism.
We will construct by introducing an -bimodule , checking that , and invoking Theorem B.
Let us recall first (cf. [20], [36]) that can be identified with -equivariant -theory of the Steinberg variety . We have the projection ; for a locally closed subvariety we will denote by .
From [62] and [10, Theorem 55, §11.3] it follows that is identified with ; the isomorphism is compatible with the action of . Here, the -action comes from the element of an triple ; we fix such a triple. It is well-known that the centralizer is a maximal reductive subgroup in , thus it is identified with .
Let be the Springer resolution and the Slodowy slice, we also let denote . This is a smooth variety called Slodowy variety. We have ; moreover, (see e.g. [44, Section 1.8]).
Now we define
Lemma 4.1.1.
Both and are torsion-free as -modules.
Proof.
It follows from Proposition 4.1.5 below that there exists an identification of left -modules, so, in particular, they are isomorphic as -modules. Let be the two-sided cell corresponding to . Recall that which is free over with a basis consisting of . We conclude that is also free over . Proposition 4.1.14 claims that there exists an isomorphism , implying that is also free over . ∎
4.1.2. The -bimodule structure on
Proposition 4.1.2.
There are natural commuting actions of and on .
Proof.
Proposition essentially follows from [43]. We sketch the proof for the reader’s convenience.
Let us define the action of on . Consider the diagram.
where , are the projections onto the corresponding factors.
Pick
and .
Now
1) can be viewed as a class of an equivariant complex on with support on ;
2) can be viewed as a class of an equivariant complex on with support on .
We define , where is the projection onto the first and the third factors. Then we set (cf. ).
The fact that this defines an action of commuting with the action of (defined below) follows by diagram chase.
Let us define the action of on
Consider the following diagram:
it is clear that for and the formula:
gives a well-defined right -action on .
∎
Lemma 4.1.3.
The induced action of on factors through the action of .
Proof.
Let us first of all recall that we have the identification . After this identification, the realization of as a subquotient of can be described as follows. We have , then the embedding is given by the pushforward for the closed embedding and the quotient is given by the restriction to the open subset .
So, our goal is to check that acts trivially on . Indeed, let be an equivariant complex representing some class in , then the support of is contained in . Let be an equivariant complex on representing a class in , then the support of is contained in . We conclude that the support of (see the notations from the proof of Proposition 4.1.2) is contained in , hence the (derived) tensor product above is equal to zero. ∎
4.1.3. Alternative description of the action of on
We can define the convolution algebra structure on by identifying it with , the -group of -equivariant coherent sheaves on with set-theoretic support on (cf. [56]). This algebra is isomorphic to (see [56, Theorem B.2]). Let us describe the action geometrically. Consider the following diagram
Since is proper, is smooth, and is smooth and proper, it is clear that for
the formula:
| (4.1.1) |
give a well-defined left -action on .
Proposition 4.1.4.
Proof.
Same argument as in the proof of [56, Theorem B.2] works. ∎
4.1.4.
In view of Theorem B, the next key proposition yields the desired homomorphism
| (4.1.2) |
Proposition 4.1.5.
We have a canonical isomorphism of left -modules: .
Remark 4.1.6.
Notice that splits as a direct sum indexed by components of , thus Proposition 4.1.5 implies such a decomposition for .
Similar direct sum decompositions are found in the literature. In particular, [30, Theorem 2.10] provides a semi-orthogonal decomposition for the derived category of equivariant coherent sheaves in a rather general situation, which implies a direct sum decomposition for the Grothendieck group. Existence of such a decomposition compatible with the convolution action would imply Proposition 4.1.5.555The similarity between Theorem 4.1.5 and results of [30] was pointed out to us by Do Kien Hoang.
On the other hand, [44, Lemma 14.9] essentially proves that for the Bialynicky-Birula filtration on can be split by means of the action of standard generators of Hecke algebra on .
We do not know if either of the two approaches yields a proof of Proposition 4.1.5.
Remark 4.1.7.
The proof of Proposition 4.1.5 below can be generalized to yield a similar direct sum decomposition for an arbitrary projective variety with a -action such that for every connected component the attractor of this component is smooth (in particular, the component itself is smooth) and the analog of Lemma 4.1.13 below holds.
Before proceeding to the proof of the Proposition we recall some geometric properties of the variety found in [26]. Recall that we fix an -triple . The adjoint action of on induces the decomposition . Let (resp. ) be the connected algebraic subgroup of whose Lie algebra is (resp. ). Recall that is the Weyl group of , let be the Weyl group of . Recall that . For we set . We also set . Note that consists of points such that , where the action of comes from the cocharacter of the center of that integrates .
Lemma 4.1.8.
Variety is smooth.
Proof.
[26, Proposition 3.2]. ∎
Warning 4.1.9.
Variety may be disconnected. It may also be empty.
Lemma 4.1.10.
The map is an affine fibration .
Proof.
Follows from [18, Theorem 4.1] (the assumptions of this theorem are satisfied for by Lemma 4.1.8 together with [54, Corollary 2 in Section 3]).
Another argument (that works only over fields of characteristic ) can be found in [26, Section 3.4], where the authors use the result of Bass-Haboush (see [26, Section 1.5]) to obtain the statement. Finally, another argument (that works over fields of arbitrary characteristic) is given in [45, Section 5]. We are grateful to George Lusztig for pointing out this reference to us. ∎
Definition 4.1.11.
A partition of a variety as a finite union of locally closed subvarieties is said to be an -partition if the subvarieties in the partition can be indexed in such a way that is closed in for .
Lemma 4.1.12.
Varieties form an -partition of .
Proof.
See [26, Section 3.4]. ∎
Proof.
(of Proposition 4.1.5) Step 1. By Lemma 4.1.12 varieties form an -partition of , i.e. there exists a labeling , of these varieties such that is closed for .
This yields a filtration (that we denote by ) on .
Moreover, the locally closed strata are the attracting varieties for , which is a union of components of . By Lemma 4.1.10 the map is a vector bundle, hence it induces (cf. loc. cit.) an isomorphism .
We claim that one gets a canonical isomorphism of -modules: .
To check this, one has to show that all of the maps are injective.
For this, see the Lemma 4.1.13 below.
In order to finish the proof of the Proposition it now suffices to provide an - and -equivariant splitting of .
Step 2. For every , together with the locally closed embedding fits into the diagram:
We claim that is locally closed inside , or, equivalently, inside . To see this, consider the diagram:
in which the first map is the graph map for which is clearly closed, and the second one is the open embedding. The claim follows.
Now let be the closure of the image of inside . It is a proper -subvariety with an open subset isomorphic to . Moreover, there are canonical -equivariant projections and from to and respectively.
Step 3. Now induction by shows that the map of the form
splits the filtration. (We recall that is a vector bundle, hence it induces an isomorphism in equivariant -theory.)
Note that is well-defined: pullback in -theory is well-defined for morphisms with smooth target or a base change of such morphisms, while is a base change of the morphism with smooth target .
Step 4. The argument similar to the one in the last paragraph shows that has a canonical left convolution action on . Moreover, and are -equivariant.
Let us, for example, prove this for .
We have:
| (4.1.3) |
We need to prove that for and the following equality holds:
| (4.1.4) |
where , (resp. ), are projections of (resp. ) to the corresponding factors.
The equality (4.1.4) is clear since the diagram (4.1.3) is Cartesian, the map is flat, and the result of [58, Proposition 1.4] holds.
This finishes the proof. ∎
Lemma 4.1.13.
The natural maps are injective for all .
Proof.
Step 1. To prove the sought-for injectivity, it suffices to construct the map
so that the maps composed with the projections are injective for all .
We will make use of the Drinfeld-Gaitsgory degeneration (cf. [25, Section 2]), and the specialization in the equivariant -theory (cf. [20, Section 5.3]).
First of all, we briefly recall both constructions.
Step 2. First, for an algebraic variety equipped with a -action, the DG-degeneration is the family so that can be canonically identified with , and . Here, for the given , stands for the corresponding attracting set, and stands for the repelling one. Moreover, there exists a canonical global trivialization of over : .
By inspection of the constructions from loc. cit., one sees that for equipped with the action of an algebraic group (so that the -action commutes with ), the family also carries an action of compatible with the trivial action on the base.
Second, if is any -equivariant algebraic family (for example, the one above), the map can be defined.
Step 3. Now, let us consider with the -action on the second factor as .
The desired map is as follows.
For any class , one may consider the class
for being the projection .
Let us now prove the fact that the composition restricted to is non-zero for a non-zero .
Let be the support of . Clearly, is contained in so it lies in the closure to the component of corresponding to . Set and let be the Drinfeld-Gaitsgory interpolation of . Note that is closed in by [25, Proposition 2.3.2 (i)]. Note also that is a union of connected components of .
Let be an evident open (and closed) embedding , and let be an evident open embedding . They glue to an open embedding . Let be the restriction of to .
Then,
| (4.1.5) | |||
| (4.1.6) | |||
| (4.1.7) |
Here in the second step we use that the embedding degenerates, via the Drinfeld-Gaitsgory family, to the embedding and take the limit for the corresponding trivial family , – and [20, Lemma 5.3.6] (cf. also [20, Theorem 5.3.9]). We have also omitted the Thom isomorphism from the notation.
The claim follows.
∎
The proof of the following fact is similar to the above discussion.
Proposition 4.1.14.
As a right -module, is isomorphic to .
Since is smooth, is a unital algebra, where the unit element is the class of the structure sheaf of the diagonal .
Corollary 4.1.15.
The map is injective.
Proof.
Let be the identification of right -modules (see Proposition 4.1.14 above). It follows from the definitions that for , we have , i.e., is uniquely determined by . The claim follows. ∎
4.2. Surjectivity of
To finish the proof of Theorem A it remains to show that is surjective. Recall that is the isomorphism of Proposition 4.1.14, it is compatible with the -action and the right -action.
Consider the identifications given, respectively, by , by Proposition 4.1.5 and by Corollary 2.3.2. Let be the image of .
Lemma 4.2.1.
The element is left invertible.
Proof.
It follows from Proposition 4.1.14 that the element generates under the right action of , hence, also under the right action of ; here we use that the -action on comes from the homomorphism . Generators of a free rank one right module are exactly right invertible elements. We conclude that there exists such that . Recall that is the complexification of . Since is left-Noetherian (being a finite module over a Noetherian central subalgebra, see e.g. [39, Proposition 1.6]), it is a Dedekind-finite ring, so the right invertible element is also left invertible. In particular, it follows that the element is the left inverse to . ∎
Corollary 4.2.2.
An element is uniquely determined by its value on .
Proof.
By Theorem B such an endomorphism is given by right multiplication by an element of , since is left invertible, the claim follows. ∎
Since is a free generator of as a right -module, Corollary 4.2.2 implies surjectivity of . This completes the proof of Theorem A, establishing the isomorphism
| (4.2.1) |
Remark 4.2.3.
As pointed out above, by Theorem A the monoidal category can be viewed as a categorification of the algebra . It would be interesting to find a compatible categorification of the homomorphism . We do not have a proposal for a monoidal functor categorifying ; however, our proof provides a categorification for the bimodule structure on the target ring arising from that homomorphism: the above argument shows this bimodule is identified with , the bimodule category is its categorification.
Corollary 4.2.4.
We have
4.3. Towards the geometric description of and
4.3.1.
It follows from the definitions that the homomorphisms
can be described as follows. Let
be the identification given by (see the notation in the proof of Step of Proposition 4.1.5).
Proposition 4.3.1.
The homomorphism is given by
| (4.3.1) |
where is the convolution action of on .
The homomorphism has the same description with being replaced by the convolution action of on .
Proof.
The homomorphism is given by the bimodule . Namely, for , its image in under is obtained as follows: we consider the operator and use the isomorphism of right -modules to identify . Then, is the element of corresponding to . In other words, is the element of such that
for every . Substituting , we conclude that . ∎
Remark 4.3.2.
Note that can be explicitly described, it is equal to the structure sheaf of the disjoint union .
4.3.2.
To every character of the maximal torus of we can associate the corresponding induced line bundle . Let be the pull back of to . Let be the diagonal embedding. The elements form the so-called ‘‘lattice part’’ of the affine Hecke algebra (c.f. [20, Section 7]).
Remark 4.3.3.
Note that the action of on is given by .
We will denote by the restriction of to . Abusing the notation, we will denote by the push forward of under the diagonal embedding .
Warning 4.3.4.
It is not true in general that for , the element coincides with .
Informally, the reason that leads to the Warning can be illustrated by the following example. Consider with the action of on it given by . Then, the set consits of two points , . The identification is induced by the correspondence . It follows from the definition that this identification is given by
where by we denote the skyscraper sheaf at the point point .
We see that
i.e., does not commute with tensoring by the line bundle .
let us now work out the details of the similar phenomenon in the real situtation, i.e., for . Let be the subregular nilpotent. Then, the Springer fiber has two irreducible components labeled by simple roots , . Each of these components is isomorphic to so we will denote the ’th component by . Components , intersect transversally at one point to be denoted . The set consists of three points, one of them is and two other are such that .
It is easy to see that the correspondence is equal to
| (4.3.2) |
Let be the projection of the correspondence (4.3.2) onto and let be the projection of (4.3.2) onto .
Let be any dominant weight, let be such that . Take (where is as in Warning 4.3.4.)
Hence, it follows from (4.3.1) that the homomorphism sends to (for the first equality see Remark 4.3.3 above)
So, our goal is to compare
Recall that , it follows that is equal to
Clearly,
when (i.e., or .)
4.4. Case
4.4.1.
Let us consider the case in more detail. We then have , the action of on is trivial. It follows that the splitting is also trivial and the correspondence is equal to (the diagonal ). Let be the two-sided cell corresponding to . Theorem A in this case establishes the isomorphism:
such that the homomorphism is given by
where corresponds to the standard action of on via convolution.
Remark 4.4.1.
The map can be explicitly described as follows. Let us consider together with its three projections onto each of the factors. Let also be the natural map
Unwrapping the definitions, one gets : here we have identified and by means of the Thom isomorphism .
Let us also consider the natural map By the projection formula, for the natural map .
The identification was already obtained in [59] (when the derived subgroup of is simply-connected) and in [49] (in general). Moreover, in [63] the homomorphism was described for such that the derived subgroup of is simply-connected. For such , is isomorphic to the matrix algebra (see [36, Proposition 1.6]), this identification is used in construction of given in [63]).
The geometric construction in loc. cit. is equivalent to the one from the Remark above (cf. Theorem 3.5 in loc. cit.) up to the conjugation by some explicit matrix.
4.4.2. Example when a specialization of at is not semisimple
It is known that is not isomorphic to a matrix algebra over in general (see [60, Section 8.3]). Actually, it is even not true that the fiber of at every point of is semisimple (c.f. [12, Corollary 1] and [55, Example 4.4]).
For example, for , and , the fiber of at is four-dimensional but the are only two irreducible representations of and both of them are one-dimensional. To see this, note that by Theorem C (see Section 5.1 and Proposition A.2.1 below), irreducible modules over are all of the form , where is an irreducible representation of acting simply transitively on . So, is two dimensional and is the direct sum of two irreducible one-dimensional -modules.
5. Representation theory of from the geometric perspective.
5.1. Simple modules over
The goal of this Section is classification of simple -modules resulting in the proof of Theorem C.
We first recall Lusztig’s classification [40, Theorem 4.2]. It is shown in loc. cit. that for every pair of a semisimple element and there exists unique irreducible -module characterized by the following property: is isomorphic to for generic . Moreover, Lusztig proved that every irreducible -module is of the form for some as above.
Our goal is to prove that:
(a) irreducible modules over are of the form ,
(b) we have: .
Part of this theorem immediately follows from Proposition A.2.1. Part is a consequence of the next Lemma. Set .
Lemma 5.1.1.
There are canonical isomorphisms of - and -modules respectively:
Proof.
We prove the claim for , the argument for follows.
We have checked (see Remark 4.2.3) that the -bimodule corresponding to the homomorphism is identified with , thus we have a canonical isomorphism:
holding for any -module .
Thus we are reduced to constructing an isomorphism:
| (5.1.1) |
Let us denote by the product of our and (Zariski closure of the cyclic subgroup generated by ). Setting , we have an isomorphism of -modules:
| (5.1.2) |
this follows from the fact that (respectively, ) is a free rank one module over (respectively, ), both sides of (5.1.2) are isomorphic to .
Consider
Here the central subalgebra acts on the right factor via the character corresponding to , we will denote reduction by that character by the subscript . By Section A.1.5 we get (cf. also (5.2.2) below):
| (5.1.3) |
where in the last equality we use [20, Theorem 5.6.1] (the implication , for the equivariance under the trivial group).
Strictly speaking, to use it, we should establish the Kunneth formula for the smooth variety . However, it immediately follows from the second of the following two identifications:
| (5.1.4) |
For the first of these two equations, note that is also a Springer fiber (for ). Thus, is isomorphic to the Chow group , cf. [26, Theorem 3.9]. Moreover, the Chow group is isomorphic to the -theory by [16, Theorem III.1 (b)]).
Now, the results of section A.1.5 also imply that
| (5.1.5) |
Thus
∎
5.2. Families of modules over and
In the previous section we proved that for any there exists an identification .
(It should be noted that for being not a root of the Poincaré polynomial of , this was already shown in [19] using the algebraic result from [61].)
Now, we will explain that this collection of isomorphisms fits into an algebraic family, using our geometric results on .
This fact is stated in [19] and used in the proof of [19, Theorem 1.8 (3)], cf. footnote 3 and section 6 below.
5.2.1.
We start by introducing certain algebraic family whose fibers are of the form
If is connected modulo the center of and has simply connected derived subgroup, then the -theory yields a family which has such fibers for all : the specialization at the maximal ideal of a semisimple conjugacy class of some , is
where the first isomorphism follows from localization theorem [28] (see Appendix A).
In general, the situation is more complicated.
Remark 5.2.1.
Let be a reductive group which is either disconnected or not simply connected. For a semisimple conjugacy class of some the specialization may be not isomorphic to . In particular, the finite group acts trivially on the former, while it may act non-trivially on the latter (cf. [12, 5.2]).
Nor do we have an isomorphism666However, we have (cf. Appendix A.1): (5.2.1) between and the invariants .
A counterexample is provided by acting on . Let be the element , let be the diagonal torus and . Then, is a set of two points. Thus, , where is a character lattice. It is a module of rank over , so has dimension (at least) two. On the other hand, is the one-dimensional space of -invariants.
5.2.2.
Being unable to define a single family with required fibers, we instead consider (following [19]) a collection of families defined for every semisimple .
We fix such , and a torus , and proceed to construct a family of modules over whose specialization to a point is isomorphic to . (The latter equality holds since the homology group is isomorphic to the Chow group ; cf. [26, Theorem 3.9] and the Chow group is isomorphic to the -theory by [16, Theorem III.1 (b)]).
Let us denote the diagonalizable group by . We set , where the subscript refers to restriction to the closed subset of .
From Section A.1.5 it follows that for any as above we have:
| (5.2.2) |
where by we mean completion at the maximal ideal of an element and denotes the identity component in .
Applying localization theorem to the torus we conclude that the specialization is identified with .
The ring acts naturally on , it is easy to see that the action on the fiber coincides with one introduced in Proposition A.2.1 (cf. also [20, 5.11.7, 5.11.10]).
Remark 5.2.2.
The module is similar to the semiperiodic module of [13]; it can be viewed as its generalization for , in the notation of loc. cit..
In a similar manner, one can form a family over with fibers and equip it with an action of . We set .
5.3.
Now, the natural goal is to obtain a ‘‘version in families’’ of Lemma 5.1.1 above, i.e. to prove that the family is isomorphic to the pullback of the family under the homomorphism .
We need the following auxiliary result.
Let us consider a natural morphism,
Lemma 5.3.1.
is an isomorphism.
Proof.
Thus, it suffices to check the claim fiberwise. Similarly to calculation (5.2.2) it follows from:
The last isomorphism is clear from the argument similar to the formulas (5.1.4). ∎
Set , .
Let us denote the algebra by .
Similarly to the proof of Lemma 5.1.1, one obtains a morphism
It is an isomorphism since it is an isomorphism on fibers and all involved modules are flat.
Now we claim that, moreover, .
Moreover, since the vector bundle (over the corresponding torus) is self-dual in a canonical way (thanks to Poincaré pairing), there exists the canonical morphism
One easily sees that, in fact, it establishes the isomorphism. Indeed, it is enough to check this fiberwise, and we reduce the statement to an elementary linear algebra observation.
We conclude:
Now, as in Lemma 5.1.1, we obtain:
Theorem 5.3.2.
As -modules,
Now the arguments in the previous chapter generalize verbatim to prove the following -equivariant isomorphism.
Theorem 5.3.3.
| (5.3.1) |
Remark 5.3.4.
This proves the version of Corollary 2.6 in [19] ‘‘in families’’.
So, Theorem D is proven.
6. Braverman-Kazhdan’s spectral description of
6.1.
We are now ready to reprove Theorem 1.8 (3) in [19] addressing the points in footnotes 3, 4 using our geometric approach to .
Let be a version of over a local field; let be a characteristic of a residue field; we assume that is large enough. (Our convention for bold letters is slightly different from the one in [19]).
Let be a Levi subgroup of . Let be an irreducible tempered representation of and let be a character of .
Then (cf. [4, p.78]) all representations of the form can be realized on the same vector space that we denote .
Let be the connected component of (recall that is the center of ). Then we obtain an action of on the trivial vector bundle over a torus .
By taking Iwahori-invariants, we obtain a family of -modules over (cf. [19, 1.2]): let us denote it by In particular, setting we get a family that we denote by (since is fixed it does not enter the notation).
6.2.
It is known that:
1) for any containing , the family can be naturaly realized as a subfamily , where is defined as pull
back of under ;
2) for any compact and a triple conjugate to , the family can be rationally identified with via the intertwining operator.
The next statement is an equivalent form of Theorem 1.8 (3) in [19].
Theorem 6.2.1.
Let , where the product is taken over all triples
as above with compact. Let be the subalgebra
consisting of elements which satisfy the following conditions:
a) does not have poles at points of families corresponding to non-strictly positive characters of Levi subgroups;
b) is compatible with 1) and 2) above.
Then .
The goal of this section is to summarize the proof as an application of our geometric description of .
6.3.
Let be the Springer fiber in . Then there is an embedding of the lowest (in the natural ‘‘Bialynicki-Birula’’ order) component of -fixed points: . We have a natural morphism
| (6.3.1) |
Note that the source of this map is a trivial family of -modules, the -multiplicity subspace for the action of on its fiber equals .
6.4.
For a finite group with an irreducible representation and any representation , we will denote the multiplicity space by We proceed to compare the geometrically defined family with .
Proposition 6.4.1.
There exists an open set , containing all non-strictly positive characters, so that over the natural map
| (6.4.1) |
induced by , is an isomorphism.
Proof.
It is enough to prove this statement fiberwise. In this form it is contained in [19, 2.2]. ∎
Corollary 6.4.2.
Proof.
It follows from Corollary 6.4.2 that embeds into
| (6.4.2) |
Moreover, by 2) in Section 6.2, and the fact that any character is conjugate to a non-strictly positive one, LHS can be replaced by .
Moreover, similarly to Theorem 5.3.3, we are reduced to showing that the following holds.
Proposition 6.4.3.
Let be the subalgebra of consisting of elements satisfying the following property.
For any pair of Levi subgroups whose Lie algebras contain and elements , ; , such that is conjugate to we have:
| (6.4.3) |
Then via the natural action map .
The proof of Proposition 6.4.3 will occupy the rest of this section.
6.5.
First we reduce to a finite product.
1) preserves a pinning of (say, , );
2) every element of is conjugate to an element in (the upper index stands for the invariants of -action).
We will denote by , and we will denote by the Levi subgroup .
From 2) above it follows that the natural projection
is injective.
6.6.
We calculate the completion for a semisimple conjugacy class . Let be a maximal torus inside . Let be and recall that is the connected component of .
As above, the map
(which is defined in the same manner as above) is injective.
Proposition 6.6.1.
For any , such that , we have
Here the left hand side is the completion of at the maximal ideal corresponding to .
Proof.
Step 1. It suffices to construct an -invariant function on such that and . Thus it is enough to prove that in the coarse quotient the image of is closed and does not contain the image of .
This follows from the following properties of such a quotient.
Step 2. Let be a possibly non-connected algebraic group. According to the results of Mohrdieck ([48]; cf. also [33]) there exists a set-theoretic lifting , so that:
1) there exists a maximal torus normalized by for any .
2) for a certain finite group we have:
∎
Combining the above statements we get the following.
Lemma 6.6.2.
The map is injective.
6.7.
Recall that is a maximal torus inside , let being the Weyl group of . Then the image of lies inside
| (6.7.1) |
Let be the augmentation ideal of . Let be the ideal generated by the image of in (under the restriction homomorphism ). Let us note that:
1) is the vector bundle over with fiber ; thus, it can be non-canonically identified with , where is ;
2) as was already stated, the completion is in the sense of -modules;
3) (cf. Remark A.1.5 for ).
Now we see that the image of lies inside .
Now it follows that the map (introduced in (6.4.3)) induces an identical isomorphism on completions (and, in particular, is surjective). Thus, Proposition 6.4.3 is proven.
Remark 6.7.1.
7. The structure of the cocenter of
7.1.
In this concluding section we will describe and for generic , where stands for the cocenter, i.e. the 0-th Hochschild homology of an associative ring. This will prove Conjecture from [12]. The main theorem is as follows.
For a reductive algebraic group let , be the corresponding commuting variety equipped with the natural (respectively, reduced) scheme structure.
We will be also interested in the categorical (coarse) quotients of these schemes:
Remark 7.1.1.
By the result of [46] the ring is reduced if the reductive group is connected. One expects that this theorem generalizes to the case of a not necessarily connected reductive group, such a generalization would imply that .
Let (where
‘‘’’ stands for ‘‘admissible’’) denote the vector space of regular functions on
satisfying the following property.
For a semisimple let denote the pull-back of under the map
from the centralizer of in to .
Then iff for any semisimple
the function is a linear combination of admissible characters of the group .
Here by an admissible character we understand the character of an irreducible
representation appearing in , (or, equivalently, in ; cf. [40, 2.8]).
Notice that an admissible representation factors
through the group of components of , thus for the function
is locally constant for every .
Theorem 7.1.2 (Theorem E).
We have a canonical isomorphism .
Corollary 7.1.3.
Proof.
The first isomorphism follows from Theorem 7.1.2 and Theorem A. The second one then follows from [12, Theorem 1] which establishes an isomorphism when is not a root of unity. ∎
Remark 7.1.4.
Another approach to constructing isomorphism (7.1.2), as well as a stronger version describing the unipotent part of the cocenter of the -adic group, will be presented in [11] (see also a related result [1]).
In that approach it is realized as a ‘‘decategorification’’ of a result of Ben Zvi-Nadler-Preygel [3] describing the trace of the affine Hecke category as coherent sheaves on commuting pairs of elements in .
The rest of this section is devoted to proofs.
7.2.
Let be a scheme, a locally free coherent sheaf on and and endomorphism. To this data one assigns a regular function ; this is a special case of the more general Hattori-Stallings trace (cf. [31], [53]). The trace is additive on short exact sequences, thus it extends to the derived category of perfect complexes. The construction is manifestly local in the fppf topology, so it extends to algebraic stacks.
Let now be an algebraic stack over a field and be the inertia stacks. We get two natural isomorphisms between the composed morphisms of stacks and , composing the first one with the inverse of the second we get an automorphism of the first composition (we denote that composition by ). Thus for the sheaf acquires a canonical automorphism . For example, if where is a scheme and an algebraic group then where . In this case the action of on the fiber of at equals the action of on the fiber of at .
Remark 7.2.1.
One expects an isomorphism , having such an isomorphism one could define the function as the image of the class under the trace map from -theory to Hochschild homology.777We thank Jakub Löwit who pointed it out to us. We were not able to find a reference for such an isomorphism, so we resorted to the above less direct construction.
Lemma 7.2.2.
Let be a scheme over a characteristic zero field and an affine algebraic group over . Let be a perfect complex on the stack and . For let be given by . Then is locally constant.
Proof.
Without loss of generality we can assume that is algebraically closed.
Let be the Zariski closure of the cyclic group in . Thus is abelian algebraic group, so it is a product of a diagonalizable and a vector group. The restriction of an equivariant coherent sheaf to carries an action of the diagonalizable group , thus splits as a direct sum of subsheaves where runs over the characters of , where acts on via . It follows that such a decomposition is also well-defined for a perfect complex. Since the Euler characteristic of a perfect complex is a locally constant function, the claim follows. ∎
We now consider , the quotient of a reductive algebraic group by the conjugation action. Thus ; applying the above construction we get a map . By Lemma 7.2.2 it lands in the space of functions satisfying the local constancy condition stated in the Lemma.
7.3.
We now proceed to prove the following reformulation of Theorem 7.1.2.
Theorem 7.3.1.
Set , let be the action map, and the projection. Consider the map
where the last arrow is the restriction to the reduced subscheme . (Compare with the definiton of the map in A.4.1.)
Then: 1. lies in the kernel of , and induces an injective map on .
2. The image of lies inside .
3. The image of equals .
We will start with the proof of , and then proceed with and .
7.4. Admissibility
7.4.1.
Recall that . For , we will denote by , where is the involution swapping factors. For as above, and , (by Lemma 7.2.2) can be considered as a function on invariant under the conjugation. Let us denote this function by . Recall that for we have
where is an embedding in of some lifting of .
Lemma 7.4.1.
Let be any irreducible representation of and let be its character. Then the invariant pairing is equal to (recall that acts on via the convolution).
Proof.
We start with a piece of notation.
Let us for consider the operator acting on . Let also stand for the Euler characteristic.
Step 1. Since (all ∗- and ∗-functors below are derived)
(here and below is always a graded trace), and
we have to prove that
Equivalently, we are interested in the following equality:
| (7.4.1) |
where stands for the graph embedding and .
(Let us recall, that we can interchange -theory and (co)homology of because of [26, Theorem 3.9] and [16, Theorem III.1 (b)].)
Step 2. Now we would like to use the results of Section A.4. Namely, Proposition A.4.1 together with the proof of Proposition A.2.1 say that
| (7.4.2) |
for being the diagonal inside , being a structure sheaf of the graph of inside , and being the natural embedding.
But, using a base change from [58, Proposition 1.4], as in the proof of Proposition 4.1.5, for the diagram
one sees that for any sheaves and on ,
| (7.4.3) |
where all of the functors are, as usual, derived (this also holds below); and is an embedding of the diagonal.
Here is an embedding of the graph of , and we use the following simple observation: for being a -action map, .
Step 3. On the other hand, by a localization theorem (cf. [20, 5.11.8] and the proof of our Proposition A.2.1),
| (7.4.5) |
where is the natural embedding.
7.4.2.
By the previous lemma, only admissible characters of may enter . This finishes the proof of the admissibility.
7.5. Injectivity
The fact that lies inside the kernel, follows from the proof of Lemma 7.4.1. However, the more geometric proof of the same fact exists (which also does not require our discussion about the reducedness above). For this, see A.4.1.
In view of Lemma 7.4.1, injectivity follows once we check the density of characters for , i.e., the statement that an element whose trace in every finite dimensional -module vanishes lies in . By [12, Theorem 1] the cocenter of is isomorphic to the cocenter of for almost all . Thus the density of characters for follows from the density of characters for together with the fact that pull back of finite dimensional modules generate the Grothendieck group of finite dimensional modules (see [39, Lemma 1.9]). ∎
7.6. Surjectivity
By the results of the previous subsections, we have a morphism , whose image lies in . We proceed to prove that in fact every element of the latter subalgebra lies inside .
We start with the following statement. Let be , set .
Consider , the spectrum of the ring of global functions on the inertia stack of . Any point of (which is reduced, since is smooth and is finite) can be interpreted as (the conjugacy class of) a pair , where Recall that we have the natural homomorphisms so is a module over .
Proposition 7.6.1.
The natural morphism , ; , becomes an isomorphism after completion at (as of -modules).
Proof.
The proof is based on the material of Appendix A.
Step 1. First of all, let us note that, by Section A.1.5, .
(Here we complete the RHS at ).
Let us note that . (Here we’ve chosen some set-theoretic lifting of to .)
Thus, it suffices to see the equality
Step 2. Let be the cover of , so that is simply connected. Functoriality of the construction of such a cover allows us to lift an automorphism of to . Let us denote the latter automorphism by .
First of all, let be a -invariant open neighbourhood of such that the composition is an isomorpism onto its image that we denote by . Let be the ideal corresponding to and let be its preimage. It follows from Proposition A.1.7 that we have an isomorphism
Moreover, results of Proposition A.3.2 show us that
Here -action on is obtained via the natural projection and regular right-left translations of the latter group on itself.
Now, let us note that the action of on is transitive with a stabilizer of being isomorphic to a finite cover of Let us denote this group by .
Then we get
here -action on differs from the natural one (i.e., from the one induced by the diagonal embedding ) by the -twisting.
Step 3.
After restricting to and completing at we obtain:
Recall now that and the composition is an isomorphism.
We conclude that .
And, hence, (since is an intersection of a diagonal and of a graph of acting on ).
∎
7.6.1.
Now we can start the proof of the surjectivity.
Let us now, first of all, recall that (cf. Section A.1) for any -variety , . In particular,
| (7.6.1) |
For a subalgebra , let us denote the image of its completion at in the RHS of (7.6.1) by .
Now, we have to prove that the map
| (7.6.2) |
defined analogously to , is surjective (recall that ).
7.6.2.
The main idea is to use the Morita equivalence to get another realization of the image of the same map.
To do this, let us denote by . This is a module over , and can be considered as -algebra.
Then, (cf. Section A.1.5) the LHS of (7.6.2) is isomorphic as an algebra to (indeed, we have a natural -equivariant homomorphism that becomes an isomorphism at the fiber at and both of the algebras are free modules over so our map must be an isomorphism, passing to -invariants we obtain the desired statement).
Lemma 7.6.2.
The algebra is Morita-equivalent to , where is an idempotent in the group algebra of corresponding to the set of -characters appearing in (and the completion should be understood in the same sense, as above).
Proof.
Indeed, recall that indecomposable projective modules over have the form , where is one of irreducible representations of . Recall now that as -module. In particular, is a projective -module.
Recall also that , so
is the quotient functor with the kernel consisting of such that . We need to check that the kernel consists of such that . To see that, it is enough to show that for any representation of and any module , we have iff This is clear since
∎
It follows from Lemma 7.6.2 that:
| (7.6.3) |
Thus, it follows that for some -invariant Zariski neighborhood of , one has
7.6.3.
It follows from the results of Baranovsky (cf. [2]) that there exists a map which induces an isomorphism
One can construct such an isomorphism (let us call it ) as the -th graded component of the following chain of equalities ( corresponds to Hochschild homology):
Indeed, we get
Thus, one sees that the map (where the first arrow is an evident surjection) can be characterized by the following condition: for any , ,
| (7.6.4) |
where is the irreducible -module corresponding to the pair (cf. proof of Proposition 7.6.3 below).
We will be interested in .
Proposition 7.6.3.
The isomorphism of Proposition 7.6.1 induces an isomorphisms between the images of and of .
Proof.
Irreducible representations of are easily seen to be in bijection with pairs , where , , are such that appears in for some irreducible representation of with .
The cocenters of two Morita-equivalent algebras are canonically isomorphic, the isomorphism is compatible with evaluating traces on the corresponding finite-dimensional representations. In view of this observation, Proposition 7.6.3 follows from the previous paragraph together with (7.6.4) and Lemma 7.4.1. ∎
7.6.4.
Let us consider a subring consisting of functions , such that at the every fiber of the projection , they are sums of the restrictions of -characters lying in the set corresponding to .
Remark 7.6.4.
Note that the notation ‘‘’’ is consistent with the previous one. Namely, our is the completion of at .
Clearly, . To finish the present subsection, it remains to show that, in fact, equality holds.
Indeed, let us take . Since , we can rewrite as for some . But then, since , one has .
Appendix A K-theoretic appendix
Here we summarize general properties of equivariant -groups used in the paper. None of this is original; almost everything is taken from [29], from [28], or from [47] (cf. below).
A.1. Completion in equivariant -theory
A.1.1.
Let be an algebraic group acting on a variety (we are not assuming that is smooth in this section, but we will actually apply the results of this section only to smooth ). Pick a semisimple element and set .
Let be the conjugacy class of . It follows from [28, Theorem 4.3] that there exists the isomorphism of completions
In particular, we have an isomorphism of fibers .
Remark A.1.1.
If is smooth, then we have the natural (forgetting the equivariance and restriction) maps:
| (A.1.1) |
Their composition induces the isomorphism of completions (see [28, Theorem 4.3 (b)]).
A.1.2.
Now we want to understand . For connected , it follows from [29, Theorem 1.1] that ; but this is not true in general, as we have already mentioned in Remark 5.2.1. Nevertheless, Graham explained to us that using the arguments from loc. cit., one can show that surjects onto ; but this would not be enough for us.
A.1.3.
First of all (following [28, Section 5.2]) we identify with as follows. Let us construct the isomorphism . Consider a -equivariant coherent sheaf on . The action of is trivial on so the -equivariant structure on induces the decomposition , where is the subsheaf of on which acts via the multiplication by .
The desired isomorphism is induced by the automorphism
| (A.1.2) |
Remark A.1.2.
The same argument (cf. loc. cit.) shows that .
A.1.4.
Now it remains to describe . We set , . Our goal is to describe . Let be the maximal ideal of . Recall that we have the restriction homomorphism and let be the ideal generated by the image of .
Remark A.1.3.
Note that is the maximal ideal in that is equal to the ideal . In particular, is equal to the ideal generated by in .
Proposition A.1.4.
We have an isomorphism
Proof.
Consider the induction and restriction functors
Functor here just sends to itself considered as -equivariant sheaf.
Functor sends to the following sheaf:
consider the projection morphism . We have an action of on given by .
Moreover, morphism is -equivariant (action of on is trivial); so is -equivariant. The action of on is free, so there exists the unique coherent -equivariant sheaf on such that its pullback to is isomorphic to .
Finally, we define , where is the natural (-equivariant) morphism sending to .
It is clear that, for any (resp., any ),
| (A.1.3) |
Note that the functors are exact so they induce maps between and . Note also, that both and are linear for the action of (the action of on is via the restriction homomorphism ), so (passing to the fiber at ) we get maps in both directions between and . We see that restriction of both , to
is just the multiplication by so we obtain the desired isomorphism. ∎
A.1.5.
Now let us note that is a finitely generated module over , and its fiber over one is (use [29, Theorem 1.1]).
Moreover, the ring is local commutative, so every flat module over it is free.
We conclude that if is flat over (that happens, for example, if is flat over ), then we have a (non canonical) -equivariant isomorphism:
| (A.1.4) |
Passing to -invariants and using Proposition A.1.4 we get the isomorphism:
| (A.1.5) |
Remark A.1.5.
The situation is similar to the one in Remark A.1.2: namely, under the same assumptions as above, one gets
| (A.1.6) |
Remark A.1.6.
Our ‘‘local flatness assumption’’ holds for and , to see this recall first that if is a maximal torus then is free over (same argument as in [44, Lemma 1.10]). Now, Let be the cover of , so that is simply connected. Let be the preimage of in . It follows from Proposition A.1.7 that we have so it is enough to check that is flat over . Recall now that by [47, 7.1] we have the isomorphism so it remains to check that is flat over (we use fpqc descent here). This again follows from the identification .
A.1.6. Passing to a covering
Suppose that is a connected reductive group. Let be the covering of so that is simply connected. Let be the (finite central) kernel of .
Suppose also that acts on some (smooth) algebraic variety , and that lies in the kernel of this action. Recall that , , and the the natural map is an étale morphism. Let be an open neighbourhood of such that the composition is an isomorphism onto the image that we denote by .
Proposition A.1.7.
| (A.1.7) |
Proof.
Let us note that the formula (A.1.2) (for being some ) gives a well-defined action of on .
Step 1. We claim that . Indeed, since any -equivariant sheaf has a direct sum decomposition , there is a direct sum decomposition ; – and . We now need to compare the restrictions , .
Step 2. Set and . Let us consider the restriction of to the open set . Note that , so our goal is to identify with .
The identification of with from (A.1.2) gives a natural -equivariant isomorphism:
where acts on the RHS via permuting the factors. It follows that is naturally isomorphic to .
Remark A.1.8.
As a corollary of Proposition A.1.7 we conclude that .
∎
A.2. Modules over convolution algebras
Let be a (possibly, disconnected) algebraic group acting on a smooth variety . Let us now assume that is reductive, and that the cycle morphisms , , are isomorphisms for every semisimple .
(This is true, for example, for (see our discussion around the formula (5.1.4).)
We also assume that is flat over in some neighbourhood of . We claim that the following corollary of the above results holds.
Proposition A.2.1.
Every simple module over the algebra is of the form for some semisimple and an irreducible .
Modules , are isomorphic iff , lie in the same conjugacy class.
Note that similar results for being a finite group go back to Lusztig.
Proof.
Step 1. Set . Note, first of all, that there exists a natural morphism defined analogously to the formula (A.1.2). One can now consider a composition of and the forgetful morphism; we will call this map .
One has ; it is easy to see that it induces a morphism .
Let be a -equivariant Thom class of a normal bundle to -fixed points inside ; let be .
Let be the conjugacy class of . One can consider a morphism
Here the first arrow is a natural restriction morphism for , the second one is analogous to , and the third one is induced by via the Kunneth formula.
We claim that is well-defined and is a homomorphism of algebras.
Step 2. To prove this, it would be enough to show that the morphism
is a well-defined homomorphism of algebras.
Step 3. Moreover, by (A.1.5), one can see that the image of is a semisimple subalgebra (where is the component group of ), and that can be identified with the morphism . ∎
A.3. Restriction of equivariance in -theory.
We start with the following standard lemma.
Lemma A.3.1.
Let be an embedding of algebraic tori (over ). Then there exists a collection of characters , such that
| (A.3.1) |
and is a torus of dimension for every .
Proof.
Let , be the character lattices of , . The embedding induces the surjection (see, for example, [57, Corollary 22.5.4 (iii)]). The kernel of is free of rank . Let be the generators of this kernel. It follows from the definitions that the equality (A.3.1) holds. It remains to note that for every , the quotient is torsion-free of rank . Indeed, let be any generators of the lattice . Let be such that . It follows from the definitions that classes of freely generate . ∎
Let be an embedding of connected reductive algebraic groups with simply connected derived subgroups.
We would like to discuss the following proposition. (The similar statement was proved in [47] under different assumptions.)
Proposition A.3.2.
Assume that acts on a smooth variety . Then, the natural restriction morphism provides an isomorphism:
| (A.3.2) |
Remark A.3.3.
Note that we do not make any assumptions concerning properness of , nor existence of an equivariant affine paving.
Proof.
Step 1. Let us reduce the situation to the case of and being tori.
Namely, let be a pair of compatible maximal tori of , .
Note that, by [47, Proposition 31], one has the natural restriction isomorphism
| (A.3.3) |
In particular, for being a Weyl group of , (here we use that , see, for example, [52, Theorem 4]) and similarly .
The assertion of the Proposition in the toric case reads:
now we deduce:
Step 2. Now we can assume that is a subtorus inside some torus .
Let us change the notation. , . Using Lemma A.3.1 and the induction on , we reduce to the case for some (primitive) nonzero character .
Lemma A.3.4.
The natural restriction map is surjective.
Proof.
The more powerful statement is proven in [47, Proposition 26]. The proof is as follows.
In fact, we have to prove that the restriction map is surjective.
Let us rewrite it. Note that the character , above turns and into -varieties. Thus, by [47, Corollaries 5, 12 and Theorem 8], we deduce that can be rewritten as:
The second arrow is a surjective restriction to an open subset (cf. [47, Corollary 27]), hence the lemma follows. ∎
Step 3. What remains is to prove injectivity of . This is contained in [47, Corollary 27].
More precisely, we can write down the following piece of the long exact sequence in -theory:
One has to prove that the leftmost map can be identified with the multiplication by .
But the latter is evident from the projection formula: cf. loc. cit. ∎
A.4. Traces in -theory
Suppose that is a smooth proper variety with the same assumptions as in Section A.2.
Then, by the Kunneth formula and Poincaré duality,
| (A.4.1) |
This gives a natural action of on . Moreover, it is well-known that the resulting algebra structure on coincides with the one defined via the convolution product.
Let stand for the Euler characteristic.
Proposition A.4.1.
For any , where stands for the structure sheaf of diagonal.
Proof.
Let us denote the linear functional by .
Since corresponds to the identity operator under (A.4.1), to show that , one has to establish that (cf. [32]):
1) where stands for the convolution;
2) .
Step 1. 1) is checked by diagram chase as follows.
Let , , and be the projections to the corresponding pairs of the arguments from . Let be the embedding of the diagonal into . Let denote a projection to a point.
Then one has:
| (A.4.2) |
where all of the functors are derived and is the involution permuting the factors in (the second equality is explained in more detail in the analogous calculation (7.4.3)).
Equality 1) is now clear.
Step 2. We have to show that the Euler characteristic of the derived self-intersection of the diagonal copy of inside is equal to the dimension of .
This follows easily from the Hodge theorem and the fact that , since is smooth (for more details see [17] and references therein). ∎
A.4.1. The map
Here we introduce a trace functor that plays a key role in the last section of the main text, and prove that it is actually a trace (commutator) functor.
We consider an action of an arbitrary algebraic group on a smooth variety .
Definition A.4.2.
Consider the diagram , where the first map is the action map, the second one is the projection.
The functor is defined to be the map given by this correspondence.
Lemma A.4.3.
For any we have a canonical isomorphism where stands for the convolution product.
Proof.
To fix ideas we first produce an isomorphism of fibers.
The fiber of at is precisely for being the embedding of the twisted diagonal into . One has:
Here we have performed a change of the variable, and have used the -equivariance of . It is easy to see that the identification of fibers as above is -equivariant.
We now present a modification of the above argument that works in families and yields the isomorphism of objects in .
Step 1. Consider the universal twisted diagonal
Variety is equipped with a natural morphism . By the definition,
where, as usual, all of the functors are derived, and stand (by a slight abuse of notation) for the corresponding pull-backs under the second projection , is the -th projection, , . (Note the similarity between this calculation and the formula (A.4.2).)
We can rewrite the last isomorphism using that is the restriction of to the diagonal: for being the diagonal embedding,
Step 2. In other words, for ,
for being as in the Step 1.
(We one more time abuse notation by writing for the corresponding pullback from to .)
We now proceed as in the fiberwise argument above.
Namely, let be the embedding , . By the -equivariance of , we get that .
Moreover, let be the automorphism . Since is the -equivariant automorphism ( acts on itself via conjugation), and ,
Since the composition is just , and is a -equivariant automorphism, so that , we are done. ∎
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