Some curious properties of the descent-to-peak map

We diagonalize a map, defined by Stembridge, that associates descents to peaks in the context of quasisymmetric functions. When restricted to peaks, it can be viewed as giving a random walk on the collection of peak sets. The stationary distribution of this walk is the distribution of peak sets over the symmetric group. When applied to geometric lattices, it gives the complete enumeration of chains of faces in any hyperplane arrangement having that lattice as its lattice of intersections, extending the classical results of Zaslavsky.