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 F-q, where q is a power of a good prime for G. We write F for the Frobenius morphism of G corresponding to the F-q-structure, so that G(F) 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 U-F-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 epsilon G(F), the variety P-u(0) of G-conjugates of P whose unipotent radical contains u. We prove that the number of Fq-rational points of P-u(0) is given by a polynomial in q with integer coefficients. Moreover, in case G is split over F-q and u is split (in the sense of T. Shoji, 1987), the coefficients of this polynomial are given by the Betti numbers of P-u(0) We also prove the analogous results for the variety P-u consisting of conjugates of P that contain u.