Semisimple conjugacy classes and classes in the Weyl group

We discuss a map θ from the semisimple conjugacy classes of a finite group GF of Lie type to the F-conjugacy classes of its Weyl group. We obtain two expressions for the number of semisimple classes mapped by θ into a given F-conjugacy class of W. The first involves distinguished coset representatives in the affine Weyl group and the second is the number of elements in the coroot lattice satisfying certain conditions. The Brauer complex plays a key role in the proof. The map θ has recently proved of interest in connection with probabilistic and combinatorial group theory.