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

Solomon’s descent algebra is used to define a family of signed measures MW,x for a finite Coxeter group W and x > 0. The measures corresponding to W of type An arise from the theory of card shuffling. Formulas for these measures are obtained and conjectured in special cases. The eigenvalues of the associated Markov chains are computed. By elementary algebraic group theory, choosing a random semisimple orbit on a Lie algebra corresponding to a finite group of Lie type G induces a measure on the conjugacy classes of the Weyl group W of G . It is conjectured that this measure on conjugacy classes is equal to the measure arising from MW,q (and further that MW,q is non-negative on all elements of W ). This conjecture is proved for all types for the identity conjugacy class of W , and is confirmed for all conjugacy classes for types An and Bn.