The Steinberg torus

For a Coxeter group W, the Coxeter complex is the simplicial complex whose faces are W-cosets of parabolic subgroups of W. For an irreducible affine Weyl group, we have a realization of the Coxeter complex as a triangulation of Euclidean space with a translational symmetry given by the coroot lattice. The quotient of an affine Coxeter complex by this (full rank) lattice is a triangulation of a torus, which we call the Steinberg torus. The are only finitely many facets in this complex, in bijection with the underlying finite Weyl group.

I will describe how the h-vectors of both Coxeter complexes and Steinberg tori have nice combinatorial interpretations in terms of descent-like statistics for the finite Coxeter (Weyl) group elements. Moreover, the h-vectors in both cases have a nonnegative expansion in the symmetric and unimodal ``gamma" basis suggested by Gal, the combinatorics here governed by peak-like permutation statistics.

This is joint work with Kevin Dilks and John Stembridge.