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

Let G G be a connected reductive algebraic group defined over the finite field F q \mathbb {F}_q , where q q is a power of a good prime for G G . We write F F for the Frobenius morphism of G G corresponding to the F q \mathbb {F}_q -structure, so that G F G^F is a finite group of Lie type. Let P P be an F F -stable parabolic subgroup of G G and let U U be the unipotent radical of P P . In this paper, we prove that the number of U F U^F -conjugacy classes in G F G^F is given by a polynomial in q q , under the assumption that the centre of G G is connected. This answers a question of J. Alperin (2006). In order to prove the result mentioned above, we consider, for unipotent u