  Solomon's descent algebra is used to define a family of signed measures
M(W,x) for a finite Coxeter group W and non-zero x. The measures corresponding
to W of types A and B are known to arise from the theory of card shuffling and
to be related to the Poincare-Birkhoff-Witt theorem and splittings of
Hochschild homology. Formulas for these measures are obtained 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^F induces a measure on the
conjugacy classes of the Weyl group W of G^F. It is conjectured that this
measure on conjugacy classes is equal to the measure arising from M(W,q) (and
further that M(W,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 W of types A, B, and C.
