Distinguished Conjugacy Classes and Elliptic Weyl Group Elements

We define and study a correspondence between the set of distinguished G0-conjugacy classes in a fixed connected component of a reductive group G (with G0 almost simple) and the set of (twisted) elliptic conjugacy classes in the Weyl group. We also prove a homogeneity property related to this correspondence. Introduction 0.1. Let k be an algebraically closed field of characteristic p ≥ 0 and let G be a (possibly disconnected) reductive algebraic group over k. Let W be the Weyl group of G. (For an algebraic group H, H denotes the identity component of H.) We view W as an indexing set for the orbits of G acting diagonally on B×B where B is the variety of Borel subgroups of G; we denote by Ow the orbit corresponding to w ∈ W . Note that W is naturally a Coxeter group; its length function is denoted by l : W → N. Let I be the set of simple reflections of W ; for any J ⊂ I let WJ be the subgroup of W generated by J . Now any δ ∈ G/G defines a group automorphism δ : W → W preserving length, by the requirement that (B,B′) ∈ Ow, g ∈ δ =⇒ (gBg−1, gB′g−1) ∈ O δ(w). The orbits of the W -action w1 : w → w−1 1 w δ(w1) on W are said to be the δconjugacy classes in W . Let W δ be the set of D-conjugacy classes in W . We say that C ∈ W δ is elliptic if for any J I such that D(J) = J we have C ∩WJ = ∅. For any C ∈ W δ let Cmin be the set of elements of C where the length function l : C → N reaches its minimum value. Let c be a G-conjugacy class of G. Let δ be the connected component of G that contains c and let C ∈ W δ be elliptic. For any w ∈ Cmin we set Bw = {(g,B) ∈ c× B; (B, gBg−1) ∈ Ow}. Note that G acts on Bw by x : (g,B) → (xgx−1, xBx−1). We write C♣c if the following condition is satisfied: for some/any w ∈ Cmin, Bw is a single G-orbit for the action above (in particular it is nonempty). The equivalence of “some” and “any” follows from [L5, 1.15(a)] (which is based on results in [GP]). Received by the editors September 13, 2013 and, in revised form, June 9, 2014. 2010 Mathematics Subject Classification. Primary 20G99. The author was supported in part by National Science Foundation grant DMS-0758262. c ©2014 American Mathematical Society 223