Duality for representations of a reductive group over a finite field, II

Abstract

The purpose of this paper is to construct a duality operation for representations of a reductuve group over a finite field. Its effect, very roughly speaking, is to interchange irreducible representations of small degree with ones of large degree (for example, the unit and Steinberg representation.) At level of characters, this operation has been also considered by Alvis [ 11, Curtis [2], and Kawanaka [4]. We shall now fix some notation. G will denote a connected reductive group defined over a finite field Fq, (IV, S) its Weyl group, and f the set of orbits of the Frobenius map on S. The subsets of .!? parametrize the classes of parabolic subgroup of G which are defined over Fq; let 9, be the class corresponding to Z c S. (Thus 9, is the class of Bore1 subgroups and S,= {G}.) We denote by G the group of F,-rational points of G and by 9, the set of parabolic subgroups in 9, with are defined over Fq. Similarly, if P E ,3, we denote by P its group of Fq-rational points and by U, the group of F,-rational points of its unipotent radical. Let K be an algebraically closed field of characteristic zero. All G-modules will be over K. Let E be a G-module. For each Z c ,.% we define