The Space of Unipotently Supported Class Functions on a Finite Reductive Group

Let G be a connected reductive algebraic group over an algebraic closure F of the finite field Fq of q elements; assume G has an Fq-structure with associated Frobenius endomorphism F and let l be a prime distinct from the characteristic of Fq. In [DLM1, §7.1] and [DLM2] we outlined a program for the determination of the irreducible Ql-characters of the finite group G , which showed that the problem may be largely reduced (by induction) to an explicit determination of the Lusztig restrictions RGM(χ) of all the irreducible characters χ of G F , for all rational Levi subgroups M of G. Here, and throughout this paper, the word “rational” means “stable under the action of F”. As shown in [DLM2], this problem may be addressed through the determination of the Lusztig restrictions R M (Γu), where Γu is the generalized Gelfand-Graev character corresponding to the G -conjugacy class of the rational unipotent element u ∈ G . Now the characters Γu are examples of class functions on G F which vanish outside the unipotent set. Such functions form a vector space over Ql, which we denote by Cuni(G F ); it is the space of unipotently supported class functions onG. The Γu form a basis of this space, and our strategy in this work will be to determine the map RGM : Cuni(G F ) → Cuni(M F ) explicitly. We shall use Lusztig’s orthogonal decomposition of the space Cuni(G F ) into summands corresponding to “rational blocks” (see below) and determine RGM on each block generically, i.e. in terms of Weyl group data which is associated with the block. In particular, we obtain a simple expression for the Lusztig restriction of generalized Green functions. We then express the generalized Gelfand-Graev characters in terms of this basis to describe their Lusztig restriction. In [DLM2] we computed RGM of the generalized Gelfand-Graev character which corresponds to a regular unipotent class. In this work, we apply the general method to carry out the corresponding computation explicitly in the subregular case. Our general result on RGM of generalized Gelfand-Graev characters (6.11) essentially reduces this computation to the two problems of finding the Poincaré polynomials P̃ι,κ of certain intersection complexes on closures of unipotent classes, and to the computation of inductionrestriction tables for twisted characters of Weyl groups. In §8 we also prove a result (8.1 below) which reduces these computations in the case of SLn to the case of GLn′ , for various n. These investigations are part of our strategy of reducing the computation of character values to the case of “high” unipotent classes in the usual partial order. The first five sections of this paper consist largely of a recasting of the of work of Lusztig, which may be found in [L],[L2],[L3], in a form which permits practical computation. They also contain several orthogonality relations for Green functions and their generalizations, which are proved by relating the inner product in Cuni(G F ) to the inner product of twisted class functions