Cellini's Descent Algebra, Dynamical Systems, and Semisimple Conjugacy Classes of Finite Groups of Lie Type

By algebraic group theory, there is a map from the semisimple conjugacy classes of a finite group of Lie type to the conjugacy classes of the Weyl group. Picking a semisimple class uniformly at random yields a probability measure on conjugacy classes of the Weyl group. We conjecture that this measure agrees with a second measure on conjugacy classes of the Weyl group induced by a construction of Cellini which uses the affine Weyl group. This is verified in some cases such as type C odd characteristic. For the identity conjugacy class in type A, the proof of the conjecture amounts to an interesting number theoretic reciprocity law. More generally the type A case leads to number theory involving Ramanujan sums. Models of card shuffling, old and new, arise naturally. In type C even characteristic connections are given with dynamical systems. We indicate, at least in type A, how to associate to a semisimple conjugacy class an element of the Weyl group, refining the map to conjugacy classes.