Descent systems for Bruhat posets

Let (W,S) be a finite Weyl group and let w ∈W. It is widely appreciated that the descent set D(w) = {s ∈ S | l(ws) < l(w)} determines a very large and important chapter in the study of Coxeter groups. In this paper we generalize some of those results to the situation of the Bruhat poset W J where J ⊂ S. Our main results here include the identification of a certain subset SJ ⊂ W J that convincingly plays the role of S ⊂ W, at least from the point of view of descent sets and related geometry. The point here is to use this resulting descent system (W J, SJ) to explicitly encode some of the geometry and combinatorics that is intrinsic to the posetW J. In particular, we arrive at the notion of an augmented poset, and we identify the combinatorially smooth subsets J ⊂ S that have special geometric significance in terms of a certain corresponding torus embedding X(J)