Semisimple Orbits of Lie Algebras and Card-Shuffling Measures on Coxeter Groups

Random walk on the chambers of hyperplane arrangements is used to define a family of card shuffling measures HW, x for a finite Coxeter group W and real x ≠ 0. By algebraic group theory, there is a map Φ 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 HW, q. The possibility of refining Φ to a map to elements of the Weyl group is discussed.