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 ∈ X and B is a Borel subgroup such that B, gBg−1 are in relative position w, then V is a homogeneous G-space. Introduction 0.1. Let G be a connected reductive algebraic group over an algebraically closed field k of characteristic p. Let W be the Weyl group of G. Let B be the variety of Borel subgroups of G. For each w ∈ W let Ow be the corresponding G-orbit in B × B. Let l : W → N be the standard length function. Let W be the set of conjugacy classes in W. For C ∈ W let dC = minw∈C l(w) and let Cmin = {w ∈ C; l(w) = dC}; let Φ(C) be the unipotent class in G associated to C in [L4, 4.1]. For any conjugacy class γ in G and any w ∈ W we set Bw = {(g,B) ∈ γ×B; (B, gBg−1) ∈ Ow}; note that G acts on Bw by x : (g,B) → (xgx−1, xBx−1). For w ∈ W let μ(w) be the dimension of the fixed point space of w : V → V where V is the reflection represention of the Coxeter group W. We say that w or its conjugacy class is elliptic if μ(w) = 0. Let Wel be the set of elliptic conjugacy classes in W. The following is the main result of this paper. Theorem 0.2. Let C ∈ Wel and let w ∈ Cmin, γ = Φ(C). Then Bw is a single G-orbit. In the case where p is not a bad prime for G, the weaker result that Bw is a union of finitely many G-orbits is already known from [L4, 5.8(a), (b)]. 0.3. In the setup of 0.2 let g ∈ γ and let B g = {B ∈ B; (B, gBg−1) ∈ Ow}. Let Z(g) be the centralizer of g in G. The following result is an immediate consequence of 0.2. (a) B g is a single orbit for the conjugation action of Z(g). It is likely that Theorem 0.2 (and its consequence (a)) continues to hold if C is a not necessarily elliptic conjugacy class. See 4.2 for a partial result in this direction. Received by the editors January 13, 2011 and, in revised form, June 17, 2011. 2010 Mathematics Subject Classification. Primary 20G99. Supported in part by the National Science Foundation. c ©2012 American Mathematical Society Reverts to public domain 28 years from publication