  Random walk on the chambers of hyperplanes arrangements is used to define a
family of card shuffling measures $H_{W,x}$ for a finite Coxeter group W and
real $x \neq 0$. By algebraic group theory, there is a map from the semisimple
orbits of the adjoint action of a finite group of Lie type on its Lie algebra
to the conjugacy classes of the Weyl group. Choosing such a semisimple orbit
uniformly at random thereby induces a probability measure on the conjugacy
classes of the Weyl group. For types A, B, and the identity conjugacy class of
W for all types, it is proved that for q very good, this measure on conjugacy
classes is equal to the measure arising from $H_{W,q}$.

Group Theory;Combinatorics