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

Abstract

AbstractLet be a simple algebraic group over an algebraically closed field and let be a graph‐automorphism of . We classify the spherical unipotent conjugacy classes in the coset . As a by‐product, we show that J.‐H. Lu's characterization in characteristic zero of spherical conjugacy classes in by the dimension formula also holds for spherical unipotent conjugacy classes in in positive characteristic. If has order 2, we provide an alternative description of the restriction to spherical unipotent conjugacy classes in , of Lusztig's map from the set of unipotent conjugacy classes in to the set of twisted conjugacy classes of the Weyl group of . We also show that a twisted conjugacy class in the Weyl group has a unique maximal length element if and only if it has maximum in the Bruhat order (a result previously proved by X. He).