On conjugacy of unipotent elements in finite groups of Lie type

Abstract

Abstract. Let G be a connected reductive algebraic group defined over F q , where q is a power of a prime p that is good for G. Let F be the Frobenius morphism associated with the F q -structure on G and set G ¼ G F , 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 ¼ P F and U ¼ U F . Let G uni be the set of unipotent elements in G. In this note we show that the number of conjugacy classes of U in G uni is given by a polynomial in q with integer coe‰cients.

The Divisibility Graph of finite groups of Lie type

“…Proof. We first note that it is well known that C G (u) admits a Levi-decomposition, see for example [22,Prop. 3.2]…”

The average size of the kernel of a matrix and orbits of linear groups

“…Using a different approach, Pak and Soffer recently provided a confirmation for . While Higman's conjecture remains open in general and despite some evidence suggesting that it may fail to hold for large (see ), it nonetheless influenced and inspired numerous results on related questions; see, in particular, work of Isaacs on character degrees of so‐called algebra groups and work of Goodwin and Röhrle on conjugacy classes of unipotent elements in groups of Lie type.…”

On the endomorphism algebra of generalised Gelfand-Graev representations

“…behaviour which is independent of the associated finite field. Recently S. Goodwin and G. Röhrle have written two papers ( [GR09a], [GR09b]) in which the notion of polynomials in q is used extensively, and the set-up in the present paper is inspired by the set-up in those papers.…”