ELLIPTIC ELEMENTS IN A WEYL GROUP: A HOMOGENEITY PROPERTY

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 is an element of X and B is a Borel subgroup such that B, g Bg(-1) are in relative position w, then V is a homogeneous G-space.