  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 conjecture is
confirmed in special cases such as type C odd characteristic and the identity
conjugacy class in type A. Models of card shuffling, old and new, arise
naturally. Type A shuffles lead to interesting number theory involving
Ramanujan sums. It is shown that a proof of our conjecture in type C even
characteristic would give an alternate solution to a problem in dynamical
systems. An idea is offered for how, at least in type A, to associate to a
semisimple conjugacy class an element of the Weyl group, refining the map to
conjugacy classes. This is confirmed for the simplest nontrivial example.

Number Theory;Dynamical Systems