Counting semisimple orbits of finite Lie algebras by genus

The adjoint action of a finite group of Lie type on its Lie algebra is studied. A simple formula is conjectured for the number of semisimple orbits of a given split genus. This conjecture, is proved for type A, and partial results are obtained for other types. For type A a probabilistic interpretation is given in terms of card shuffling. (C) 1999 Academic Press.