Spherical conjugacy classes and involutions in the Weyl group

Let G be a simple algebraic group over an algebraically closed field of characteristic zero or positive odd, good characteristic. Let B be a Borel subgroup of G. We show that the spherical conjugacy classes of G intersect only the double cosets of B in G corresponding to involutions in the Weyl group of G. This result is used in order to prove that for a spherical conjugacy class O with dense B-orbit v(0). BwB there holds l(w) + rk(1-w) = dim O extending to the case of groups over fields of odd, good characteristic a characterization of spherical conjugacy classes obtained by Cantarini, Costantini and the author.