On the generalised Springer correspondence for groups of type

By Jonas Hetz

Abstract

We complete the determination of the generalised Springer correspondence for connected reductive algebraic groups, by proving a conjecture of Lusztig on the last open cases which occur for groups of type .

1. Introduction

Let be a connected reductive algebraic group over an algebraic closure of the finite field with elements, where is a prime. Let be the Weyl group of and be the set of all pairs where is a unipotent conjugacy class and is an irreducible local system on (taken up to isomorphism) which is equivariant for the conjugation action of . The Springer correspondence (originally defined in Reference 39 for not too small; for arbitrary see Reference 15) defines an injective map which plays a crucial role, for example, in the determination of the values of the Deligne–Lusztig Green functions of Reference 5; for recent surveys see Reference 7, Chap. 13 and Reference 10, §2.8. However, the map is not surjective in general. In order to understand the missing pairs in , Lusztig Reference 17 developed a generalisation of Springer’s correspondence. This in turn constitutes a substantial part of the general problem of computing the complete character tables of finite groups of Lie type.

With very few exceptions, the problem of determining explicitly the generalised Springer correspondence has been solved in Reference 17, Reference 32, Reference 38 (see also the further references there). The exceptions occur for of type with and with . Recently, Lusztig Reference 31 settled the case where is of type and stated a conjecture concerning the last open cases in type . It is the purpose of this paper to prove that conjecture, thus completing the determination of the generalised Springer correspondence in all cases.

The paper is organised as follows. In Section 2, we recall the definition of the generalised Springer correspondence, due to Reference 17. In Section 3, we explain two parametrisations of unipotent characters and unipotent character sheaves, one in terms of Lusztig’s Fourier transform matrices associated to families in the Weyl group, and the other one in terms of Harish-Chandra series, for simple groups of adjoint type with a split rational structure. In the last two sections, we focus on the specific case where is simple of type in characteristic : Section 4 contains the description of the generalised Springer correspondence for this group, so that we can formulate Lusztig’s conjecture on the last open cases as indicated above. Finally, the proof of this conjecture is given in Section 5. It is based on considering the Hecke algebra associated to the finite group (where for some ) and its natural , exploiting a well-known formula relating characters of this Hecke algebra with the unipotent principal series characters of . The fact that this formula carries subtle geometric information has been used before, for example in Reference 28.

Notation.

As soon as a prime is fixed in a given setting, we denote by an algebraic closure of the finite field with elements; furthermore, we tacitly assume to have fixed another prime , as well as an algebraic closure of the field of -adic numbers. It will be convenient to assume the existence of an isomorphism and to fix such an isomorphism once and for all⁠Footnote1, so that we can speak of “complex” conjugation or absolute values for elements of using this isomorphism. In this way, we will also identify the rational numbers or the real numbers as subsets of and just write . In several places in this paper, we will need to fix a square root of (or of powers of ), so we do this right away:

1

Strictly speaking, the existence of such an isomorphism requires the axiom of choice. However, what we really need is an isomorphism between algebraic closures of in and , and such an isomorphism is known to exist without reference to the axiom of choice, cf. Reference 4, Rem. 1.2.11.

For any finite group , we denote by the set of class functions and by the subset consisting of irreducible characters of over . Thus, is an orthonormal basis of with respect to the scalar product

Now let be a prime and be a connected reductive group over , defined over the finite subfield where for some , with corresponding Frobenius map . With respect to this , we fix a maximally split torus and an -stable Borel subgroup such that . Let be the set of roots of with respect to , and let be the subset of simple roots determined by . We denote by the Weyl group of (relative to ). Given any closed subgroup (including the case where ), we denote by its identity component, by the centre of and by the (closed) subvariety consisting of all unipotent elements of . If , we set

a finite subgroup of . In particular, is the finite group of Lie type associated to .

2. The generalised Springer correspondence

The notation is as in Section 1. In this section, we give the definition of the generalised Springer correspondence for , due to Lusztig Reference 17.

2.1.

We start by briefly introducing some of the most important notions of Lusztig’s theory of character sheaves Reference 24 and the underlying theory of perverse sheaves Reference 1. For the details we refer to Reference 17, Reference 18Reference 22. Recall that is a fixed prime. Let be the bounded derived category of constructible -sheaves on in the sense of Beilinson–Bernstein–Deligne Reference 1. To each and each is associated the th cohomology sheaf , whose stalks (for ) are finite-dimensional -vector spaces. The support of such a is defined as

(where the bar stands for the Zariski closure, here in ). We set

Let be the full subcategory of consisting of the perverse sheaves on ; the category is abelian. Assume that is a locally closed subvariety of and that is an irreducible -local system on . (We will just speak of a “local system” when we mean a -local system from now on.) There is a unique extension of to , namely, the intersection cohomology complex (where denotes shift), due to Deligne–Goresky–MacPherson (see Reference 12, Reference 1). The following notation will be convenient: For and as above, and for any closed subvariety such that , we denote by

the extension of to , by on . Now let us bring the Frobenius endomorphism into the picture. Let be its inverse image functor. Let be such that in , and let be an isomorphism. For each and , induces linear maps

Since is non-zero only for finitely many , one can define a characteristic function Reference 19, 8.4

In Reference 18, §2, Lusztig defines the character sheaves on as certain simple objects of which are equivariant for the conjugation action of on itself. We denote by a set of representatives for the isomorphism classes of character sheaves on . An important subset of is the one consisting of cuspidal character sheaves, as defined in Reference 18, 3.10. We denote by a set of representatives for the isomorphism classes of cuspidal character sheaves on . The inverse image functor may also be regarded as a functor , and we have . Thus, we can consider the subset

consisting of the -stable character sheaves in . Of particular relevance for our purposes will be the set of (representatives for the isomorphism classes of) unipotent character sheaves. These are, by definition, the simple constituents of a perverse cohomology sheaf for some and some , where is as defined in Reference 18, 2.4, with respect to the trivial local system on the torus .

2.2.

Let be the set of all pairs where is a unipotent conjugacy class and is an irreducible local system on (up to isomorphism) which is equivariant for the conjugation action of on . Let us consider a triple consisting of a Levi complement of some parabolic subgroup of , a unipotent class of and an irreducible local system on (up to isomorphism) which is equivariant for the conjugation action of on ; furthermore, assume that is a cuspidal pair for in the sense of Reference 17, 2.4, where denotes the inverse image of under the canonical map . The group acts naturally on the set of all such triples by means of -conjugacy:

where is the inner automorphism given by conjugation with . Denote by the set of equivalence classes of triples as above under this action; however, by a slight abuse of notation, we will still just write rather than or the like. We typically write for elements of and for elements of . By Reference 17, §6, any gives rise to a certain perverse sheaf , whose endomorphism algebra is isomorphic to the group algebra of the relative Weyl group , see Reference 17, Thm. 9.2. Thus, the isomorphism classes of the simple direct summands of are naturally parametrised by , so we have

where is the simple direct summand of corresponding to , and where . Then, for any , there exists a unique for which

and the isomorphism class of is uniquely determined by this property among the simple perverse sheaves which are constituents of Reference 22, 24.1. So for each , the above procedure gives rise to an injective map

Conversely, given any , there exists a unique such that is in the image of the map just defined. So there is an associated surjective map

whose fibres are called the blocks of . For any , the elements in the block are thus parametrised by the irreducible characters of . The collection of the bijections

is called the generalised Springer correspondence. Given a pair and corresponding to under Equation 2.2.3, we will write . Considering the element , map Equation 2.2.3 defines an injection

which is called the (ordinary) Springer correspondence. The problem of determining the generalised Springer correspondence (that is, explicitly describing bijections Equation 2.2.3 for all ) can be reduced to considering simple algebraic groups of simply connected type, thus can be approached by means of a case-by-case analysis. This has been accomplished for almost all such , thanks to the work of Lusztig Reference 17, Lusztig–Spaltenstein Reference 32, Spaltenstein Reference 38 (see also the references there for earlier results concerning the ordinary Springer correspondence), and again Lusztig Reference 31, the only remaining open problems occur for of type in characteristic (for which a conjecture is made in Reference 31, §6). In particular, the ordinary Springer correspondence Equation 2.2.4 is explicitly known in complete generality.

2.3.

We keep the setting of 2.2 and consider the Frobenius map . It defines actions on and , given by

and

Let , be the respective sets of fixed points under these actions where, in terms of the local systems, this is only meant up to isomorphism, and for the triples in in addition only up to -conjugacy. The map commutes with the action of on , , so it gives rise to a surjective map . Furthermore, the generalised Springer correspondence Equation 2.2.3 induces bijections

(Here, we denote by the subset consisting of all irreducible characters of which are invariant under the automorphism of induced by .) Let and , and assume that is the corresponding element of . Choosing an isomorphism which induces a map of finite order at the stalk of at any element of allows the definition of a unique isomorphism (see Reference 22, 24.2 and also 3.11). We set

We then have

so we can define an isomorphism by the requirement that is equal to the isomorphism induced by . It is shown in Reference 22, (24.2.4) that for any , the induced map on the stalk of at is of finite order. Now consider the two functions

defined by

and

for . Both and are invariant under the conjugation action of on .

Theorem 2.4 (Lusztig Reference 22, §24).

In the setting of 2.3, the following hold.

(a)

The functions , , form a basis of the vector space consisting of all functions which are invariant under -conjugacy.

(b)

There is a system of equations

for some uniquely determined .

Proof.

See Reference 22, (24.2.7) and (24.2.9). Note that the restrictions Reference 22, (23.0.1) on the characteristic of can be removed, thanks to the remarks in Reference 28, 3.10.

2.5.

As an immediate consequence of Theorem 2.4 (and of 2.3), we see that:

(i)

We have for all .

(ii)

If , then implies and .

(iii)

If belong to different blocks, we have .

Let us fix any total order on such that for , we have

(Note that the latter defines a partial order on the set of unipotent classes of .) Then the matrix has upper unitriangular shape with respect to . In Reference 22, §24, Lusztig provides an algorithm for computing this matrix , which entirely relies on combinatorial data. This algorithm is implemented in Michel’s development version of CHEVIE Reference 33 and is accessible via the functions UnipotentClasses and ICCTable.