Elliptic elements in a Weyl group: a homogeneity property

Abstract

Let G be a reductive group over an algebraically closed field whose characteristic is not a bad prime for G. Let w be an elliptic element of the Weyl group which has minimum length in its conjugacy class. We show that there exists a unique unipotent class X in G such that the following holds: if V is the variety of pairs (g,B) where g\in X and B is a Borel subgroup such that B,gBg[superscript -1] are in relative position w, then V is a homogeneous G-space.