A classification of spherical conjugacy classes

Abstract

Let G be a simple algebraic group over an algebraically closed field k. We complete the classification of the spherical conjugacy classes of G begun by Carnovale (Pacific J. Math. 245 (2010), 25–45) and the author (Trans. Amer. Math. Soc. 364 (2012), 1997–2019)

A Katsylo theorem for sheets of spherical conjugacy classes

“…However the proofs of Lemmata 4.6, 4.7, 4.8 and Theorems 2.7 and 4.4 therein are still valid for groups of type A n in characteristic 2 because also in this case spherical conjugacy classes meet only Bruhat cells corresponding to involutions in the Weyl group. This follows from [9,Theorem 3.4] for unipotent classes and, in the general case, from results in [10].…”

On Lusztig's map for spherical unipotent conjugacy classes in disconnected groups

“…It has been shown in [4, 5, 12] that spherical conjugacy classes in $$G$$ may be characterized (in any characteristic) by means of a dimension formula involving the maximal Weyl group element $$w$$ for which $$BwB$$ meets a class. More precisely, let us define, for a conjugacy class $${\cal O}$$ in $$G$$, the element $$w_{\cal O}\in W$$ as the unique element in $$W$$ for which $$Bw_{\cal O}B\cap {\cal O}$$ is Zariski dense in $${\cal O}$$.…”