On the restriction of cuspidal representations to unipotent elements

Abstract

Let G be a connected split reductive group defined over a finite field F_q, and G(F_q) the group of F_q-rational points of G. For each maximal torus T of G defined over F_q and a complex linear character θ of T(F_q), let R^G_T(θ) be the generalized representation of G(F_q) defined in [DL]. It can be seen that the conjugacy classes in the Weyl group W of G are in one-to-one correspondence with the conjugacy classes of maximal tori defined over F_q in G ([C1, 3·3·3]). Let c be the Coxeter conjugacy class of W, and let T_c be the corresponding maximal torus. Then by [DL] we know that π_θ= (-1)ⁿR^G_Tc(θ ) (where n is the semisimple rank of G and θ is a character in 'general position') is an irreducible cuspidal representation of G(F_q). The results of this paper generalize the pattern about the dimensions of cuspidal representations of GL(n, F_q) as an alternating sum of the dimensions of certain irreducible representations of GL(n, F_q) appearing in the space of functions on the flag variety of GL(n, F_q) as shown in the table below.