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.