Let be a connected, reductive algebraic group over an algebraically closed field of zero or good and odd characteristic. We characterize spherical conjugacy classes in as those intersecting only Bruhat cells in corresponding to involutions in the Weyl group of .
This note presents the study of the conjugacy classes of elements of some given finite order in the Cremona group of the plane. In particular, it is shown that the number of conjugacy classes is infinite if is even, or , and that it is equal to (respectively ) if (respectively if ) and to for all remaining odd orders. Some precise representative elements of the classes are given.