Spectral Properties of Descent Algebra Elements

The descent algebra of finite Coxeter groups is studied by many famous mathematicians like Bergeron, Brown, Howlett, or Reutenauer. Blessenohl, Hohlweg, and Schocker, for example, proved a symmetry property of the descent algebra, when it is linked to the representation theory of its Coxeter group. The interest is particularly showed for the descent algebra of symmetric group. Thibon determined the eigenvalues and their multiplicities of the action on the group algebra of symmetric group of the descent algebra element, which is the sum over all permutations weighted by q^maj. And even the author diagonalized the matrix of the action of the descent algebra element, which is the sum over all permutations weighted by the new introduced statistic desX. In this article, we give a more general result by determining the eigenvalues and their multiplicities of the action on the group algebra of finite Coxeter group of an element of its descent algebra.