On Conjugacy of Unipotent Elements in Finite Groups of Lie Type

Let G be a connected reductive algebraic group defined over Fq, where q is a power of a prime p that is good for G. Let F be the Frobenius morphism associated with the Fq-structure on G and set G = G , the fixed point subgroup of F . Let P be an F -stable parabolic subgroup of G and let U be the unipotent radical of P; set P = P and U = U . Let Guni be the set of unipotent elements in G. In this note we show that the number of conjugacy classes of U in Guni is given by a polynomial in q with integer coefficients.