Rational Points on Generalized Flag Varieties and Unipotent Conjugacy in Finite Groups of Lie Type

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 let U be 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 (2006). In order to prove the result mentioned above, we consider, for unipotent u ∈ GF , the variety P0 u of G-conjugates of P whose unipotent radical contains u. We prove that the number of Fq-rational points of P0 u 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 T. Shoji, 1987), the coefficients of this polynomial are given by the Betti numbers of P0 u. We also prove the analogous results for the variety Pu consisting of conjugates of P that contain u.