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

Let 𝐺 be a simple algebraic group over an algebraically closed field and let 𝜃 be a graph-automorphism of 𝐺.We classify the spherical unipotent conjugacy classes in the coset 𝐺𝜃. As a by-product, we show that J.-H. Lu’s characterization in characteristic zero of spherical conjugacy classes in 𝐺𝜃 by the dimension formula also holds for spherical unipotent conjugacy classes in 𝐺𝜃 in positive characteristic. If 𝜃 has order 2, we provide an alternative description of the restriction to spherical unipotent conjugacy classes in 𝐺𝜃, of Lusztig’s map Ψ from the set of unipotent conjugacy classes in 𝐺𝜃 to the set of twisted conjugacy classes of the Weyl group of 𝐺. 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).