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

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