# Notes on a conjecture of Braverman-Kazhdan

###### Abstract

Given a connected reductive algebraic group defined over a finite field together with a representation of the dual group of (in the sense of Deligne-Lusztig), Braverman-Kazhdan [BK] defined an exotic Fourier operator on the space of complex valued functions on . In these notes we give an explicit formula for the Fourier kernel and a geometrical interpretation of this formula (as conjectured by Braverman and Kazhdan under some assumption on ).

all

###### Contents

## 1 Introduction

Let be a connected reductive algebraic groups over with geometric Frobenius . Fix an -stable maximal torus of and denote by the Weyl group with respect to . Let be a connected reductive algebraic group over with geometric Frobenius such that and are in duality (in the sense of Deligne-Lusztig [DL]). Lusztig defined a partition of the set of all irreducible -characters of (with a prime not dividing ) whose parts are parametrized by the -stable semisimple conjugacy classes of . We call the parts of this partition the *Lusztig series* and we denote by the set of Lusztig series of .

Let and be an other pair of connected reductive groups in duality. We assume given a morphism which commutes with Frobenius. It induces a map as explained in §3.2.

Denote by the set of irreducible characters of . It is equivalent to give oneself a function or a central function (see LABEL:defgamma for the precise relationship). Say that is *admissible* if it is constant on Lusztig series.

Using the morphism , we can transfer an admissible function into an admissible function on by putting

with

In the first part of these notes we give an explicit formula for the central function associated to .

We denote by the centralizer of in . It is an -stable Levi factor of some parabolic subgroup of . The morphism induces a group homomorphism , , where stands for .

The image of the map being a normal subgroup of , we can define a dual map which is compatible with the action of . In particular for each , we have a morphism .

For an -stable Levi factor of some parabolic of , we denote by (resp. ) the Lusztig induction (resp. restriction) on -valued central functions.

Our first main result is (see Theorem LABEL:maintheo):

###### Theorem 1.0.1.

We have

(1.1) |

where .

For the second part of these notes we assume for simplicity that (equipped with the standard Frobenius), and for some fixed non trivial additive character of where is the restriction of the trace map to invertible matrices.

We study a geometrical realization of following Braverman-Kazhdan [BK]. However, unlike Braverman-Kazhdan we do not make any assumption on .

Consider the Artin-Schreier sheaf on and we put . We then consider the complex

in the “derived category” of constructible -adic sheaves on , where is the unipotent radical of some Borel subgroup of . This complex is naturally -equivariant and comes with a natural Weil structure .

We then consider the -invariant part of the induced complex . It comes also with a natural Weil structure .

Our second main result is:

###### Theorem 1.0.2.

We have

where is the characteristic function of .

Under some assumption on (see Remark LABEL:remark), the complex was shown to be a smooth -adic sheaf on by Chen and Ngô [NC]. In this case the complex is thus a simple perverse sheaf on and the above theorem was conjectured by Braverman and Kazhdan in [BK] (and proved by them in some particular cases, for instance when is a general linear group).

## 2 Notation

###### 2.1.

Our base field is a finite field of characteristic with elements. We fix an algebraic closure and for any positive integer we let be the unique subfield of with elements. If is a tori defined over with Frobenius , we have the multiplicative norm map , .

We will need to use -adic cohomology. We therefore fix a prime different from and an algebraic closure of the field of -adic numbers. Unless specified, the letter will denote either the field of . We choose an involution , such that if is root of unity.

###### 2.2.

For any finite set , we denote by the vector space of -valued functions on . For two functions we put

For any subset of , we denote by the characteristic function of , namely the function that takes the value on and elsewhere. If is map between finite sets, then it induces , and , , and for and we have the adjunction property

Let be a finite group. We denote for the subspace of functions in which are constant on conjugacy classes and for we put

Let be another finite group. If both and acts on the same a vector space and if these two actions commute, we have an action of on . In the particular case where , we denote by the -module with underlying vector space and with twisted -action given by

for all , . Also we will denote by , , and by , .

We denote by the category of representation of on finite dimensional -vector spaces. The set of irreducible character of will be denoted by and, for each , we choose an irreducible representation with character . We assume that our choices are made such that for all , where denotes the dual character of and the dual representation of . The trivial character will be denoted by .

The action of a finite group on itself by left and right translation endows with a structure of -module given by

for all , and .

An element of defines a function as

for all .

###### Remark 2.2.1.

If is a basis of with dual basis of , then the function

is the character .

We have an isomorphism of -modules

under which corresponds to the subspace of generated by .

###### 2.3.

Let be a group. We denote by the center of , for any we denote by the centralizer of in and for a subgroup of we put

where denotes the normalizer of in . For , will denote an arbitrary representative of in .

We denote by the space of matrices over , by the subspace of upper triangular matrices, by the subspace of upper triangular nilpotent matrices and by the subspace of diagonal matrices. The groups of invertible elements of , and are respectively denoted by , and , and we denote by the upper triangular unipotent matrices. The Frobenius that raises coefficients of matrices to their -th power is called the *standard Frobenius*.

A Levi factor of a parabolic subgroup of is -conjugate to for some positive integers such that . Note that the symmetric group in letters acts on each as . Therefore we have an action of on and

and so .

## 3 Representations of finite reductive groups

### 3.1 Finite reductive groups

We fix once for all an isomorphism and an embedding .

Let be a connected reductive algebraic group over with a geometric Frobenius associated with some -structure on and denote by , the Lang map. For a maximal torus of we denote by the character group and by the co-character group. When the maximal torus is -stable, we have an action of the Galois group on . The action of the element of is given by the following formula

for all and . It permutes the roots. The transpose automorphism induces an action on and permutes the coroots.

Given an -stable maximal torus of , the -conjugacy classes of the -stable maximal tori of are parametrized by the set of -conjugacy classes of . More precisely, we let be such that . Then is an -stable maximal torus of . All -stable maximal torus of are obtained in this way for some and as above, and two maximal tori and are -conjugate if and only if and are -conjugate. The pair is then isomorphic to the pair where is the Frobenius on .

Similarly, if we fix an -stable Levi factor of some parabolic subgroup of , then the above construction shows that the -conjugacy classes of -stable Levi factors (of some parabolic subgroup of ) that are conjugate under to are parametrized by .

We say that an -stable maximal torus of is *maximally split* if it is contained in some -stable Borel subgroup of .

We will often use the notation for a maximally split -stable maximal torus of and instead of .

We will sometimes regard the elements of as functions on .

Duality

Let be another connected reductive group endowed with a Frobenius . Let be a maximally split -stable maximal torus of , an -stable Borel subgroup containing and let the Weyl group of with respect to . If there exists an isomorphism which takes simple roots (with respect to ) to simple coroots (with respect to ) and which is compatible with the action of the Galois group , then we say that and are *dual groups* [DL, Definition 5.21]. In particular, for two tori and with Frobenius and , the pairs and are dual if there exists an isomorphism compatible with Galois group actions.

The classification of connected reductive groups in terms of root data ensures the existence of a dual group for (unique up to isomorphism). From now will denote a group in duality with .

As the functor is contravariant and the functor covariant, we have a canonical anti-isomorphism such that for all and , we have

where .

It also satisfies

for all .

The map defines a bijection and so a bijection between the set of -conjugacy classes of -stable maximal tori of and the set of -stable maximal tori of . If is an -stable maximal torus of and an -stable maximal torus of which correspond to each other under the above bijection, then and are in duality.

Let be an -stable Levi factor of some parabolic subgroup of and be an -stable maximal torus of maximally split. There exists an -stable Levi factor of some parabolic subgroup of together with an -stable maximal torus of maximally split such that is dual to . More precisely, if is of the form for some then is of the form .

We have the following proposition [DL, (5.21.5)].

###### Proposition 3.1.1.

(i) There exists a bijective correspondence between the set of -conjugacy classes of pairs , where is an -stable maximal torus of and , and the set of -conjugacy classes of pairs , where is an -stable maximal torus of and .

(ii) If and are dual tori, then .

The correspondence (i) and the isomorphism (ii) of the proposition depends on the choice of the isomorphism and the embedding . Indeed, to construct the correspondence between characters and -stable points of tori from an isomorphism , we relate characters of with and -stable points of with as follows. Given a torus with a Frobenius , the choice of an isomorphism defines a surjective group homomorphism , where is such that is split over and is the -th root of unity corresponding to . The choice of the embedding defines a surjective group homomorphism by restricting a character to .

Lusztig induction

Let be an -stable Levi subgroup of some parabolic subgroup of (which may not be -stable) and denote by the unipotent radical of . The variety is equipped with an action of by left mutliplication and with an action of by right multiplication. These actions induce actions of and on the compactly supported -adic cohomology groups .

For any irreducible character of , the -vector space is thus a -module and we denote by the virtual character of defined by

for all .

The map extends linearly to an operator . Explicitely [DM, Proposition 11.2]

(3.1) |

for all and , and where .

Recall that the operator does not depend on the choice of the parabolic subgroup having as a Levi factor and so from now we will denote simply by this operator.

We have the following basic properties.

###### Proposition 3.1.2.

(i) If is an inclusion of Levi subgroups, then .

(ii) If is a character of , then .

We have the following important theorem.

###### Theorem 3.1.3 (Deligne-Lusztig).

For any irreducible character of , there exists an -stable maximal torus of together with a linear character such that

We define the Lusztig restriction by the formula [DM, Proposition 11.2]

for all and .

The two operators and are adjunct to each other with respect to .

Lusztig series

If is an -stable maximal torus of and , we call a *Deligne-Lusztig pair* of (DL pair of for short). We say that two DL pairs and of are *geometrically conjugate* if there exists some positive integer , some such that

for all where is the multiplicative norm.

We have the following proposition [DL, Proposition 5.22].

###### Proposition 3.1.4.

Geometric conjugacy classes of DL pairs are in one-to-one correspondence with -stable conjugacy classes of semi-simple elements of .

When the -conjugacy class of the DL pair corresponds to the -conjugacy class of a pair (see Proposition 3.1.1(i)) we will write sometimes instead of .

###### Theorem 3.1.5 (Deligne-Lusztig).

and have no commun irreducible constituent unless and are -conjugate.

A *geometric Lusztig series* (or *Lusztig series* for short) of associated to the geometric conjugacy class of some DL pair of is the set of all irreducible characters of which appear non-trivially in some where is geometrically conjugate to . Thanks to Theorem 3.1.3, any irreducible character of belongs to a Lusztig series and thanks to Theorem 3.1.5, the Lusztig series are disjoint and so form a partition of which is parametrized by the -stable semisimple conjugacy classes of .

Let be a DL pair of and be a corresponding semisimple element. We denote either by or the Lusztig series associated with the geometric conjugacy class of . For we will also denote by the Lusztig series which contains .

We denote by the set of Lusztig series of .

###### Proposition 3.1.6.

Let be an -stable Levi factor of some parabolic subgroup of and let . Then any irreducible constituent of belongs to . Therefore the functor induces a map .

###### Proof.

Let and be a parabolic subgroup and Borel subgroup of such that and consider the Borel subgroup of . Assume that is an irreducible constituent of . Then is an irreducible constituent of some . On the other hand, for all non-negative integer , we have (transitivity of Lusztig induction)

Therefore any irreducible consituent of appears in some for some . Note that a priori could also appear in other cohomology groups and we could have cancellation in . The proposition 13.3 of [DM] says that appears at least in some with in the geometric conjugacy class of . ∎

A morphism of tori induces a morphism between the co-character groups and so a map between the character groups of the dual tori. Since the contravariant functor is fully faithful, we get a morphism . If commutes with Frobenius then so does .

###### Remark 3.1.7.

In the case , the morphism is constructed as follows. The morphism is of the form for some cocharacters of . Regarding now the ’s as characters of via the isomorphism , we obtain , .

More generally, consider a morphism of connected reductive algebraic groups defined over which is *normal* (i.e. the image of is a normal subgroup of ).

###### Proposition 3.1.8.

There exists a normal morphism defined over which extends any morphism obtained by duality from the restriction of to maximal tori.

###### Proof.

To see this, we are reduced to the case where is a surjective morphism or the inclusion of a closed connected normal subgroup. First of all recall that any connected reductive group is the almost-direct product of the connected component of its center and a finite number of quasi-simple groups , i.e. the product map is an isogeny (that is a surjective homomorphism with finite kernel). Therefore if is a closed connected normal subgroup of , then there exists a closed connected normal subgroup of such that the is the almost-direct product of and . The isogeny induces an isogeny between the root data and so an isogeny between the dual root data. By the isogeny theorem (see for instance [Milne, Theorem 23.9]) we thus get an isogeny . Composing this isogeny with the projection we get the required morphism .

We now assume that is surjective and denote by the kernel of . As factorizes through the isogeny , we may assume that is connected. By the above discussion, the inclusion induces a surjective morphism . Also if denotes a maximal torus of , and , then we have an exact sequence of tori and so an exact sequence . Therefore is connected from which we deduce that is also connected. The map induces an isomorphism between the root data of and and so extends to an isomorphism by the isogeny theorem.

∎

###### Proposition 3.1.9.

The pull back functor , of induces a map between the sets of Lusztig series. More precisely, if , with , then any irreducible constituent of belongs to .

###### Proof.

The statement is clear if both and are direct products of a torus by a quasi-simple group. Since both and are such direct products up to central isogeny we are reduced to prove the proposition for a central isogeny. Let be an -stable maximal torus of a Borel subgroup of and let be the image of by with . By base change, the map induces a finite surjective map

Via , the groups and act on and is invariant under these actions. We thus get for all an inclusion of -modules. Expressing any in the form for some yields an -equivariant isomorphism . Therefore if and is a representation of appearing in , the -submodule of on which acts by , then appears in .

∎

### 3.2 Functoriality

Let and be two connected reductive algebraic groups with Frobenius and let be an algebraic morphism which commutes with Frobenius . The *functoriality principle* predicts a map from certain “packets” of irreducible representations of to “packets” of irreducible representations of