Rational points on generalized flag varieties and unipotent conjugacy in finite groups of Lie type

Abstract

Let G be a connected reductive algebraic group defined over the finite field Fq, where q is a power of a good prime for G. We write F for the Frobenius morphism of G corresponding to the Fq-structure, so that GF is a finite group of Lie type. Let P be an F-stable parabolic subgroup of G and U the unipotent radical of P. In this paper, we prove that the number of UF -conjugacy classes in GF is given by a polynomial in q, under the assumption that the centre of G is connected. This answers a question of J. Alperin in [1]. In order to prove the result mentioned above, we consider, for unipotent u ? GF , the variety P0u of G-conjugates of P whose unipotent radical contains u. We prove that the number of Fq-rational points of P0u is given by a polynomial in q with integer coefficients. Moreover, in case G is split over Fq and u is split (in the sense of [22, §5]), the coefficients of this polynomial are given by the Betti numbers of P0u. We also prove the analogous results for the variety Pu consisting of conjugates of P that contain u.