The Betti Numbers of Simple Embeddings

Abstract

Let G be a simple algebraic group with Weyl group (W, S) and let w ∈ W. We consider the descent set D(w) = { s ∈ S  |  l(ws) < l(w)}. This has been generalized to the situation of the Bruhat poset W ^J, where J ⊂ S. To do this one identifies a certain subset S ^J ⊂ W ^J that plays the role of S ⊂ W in the well known case J = ∅. One ends up with the descent system (W ^J, S ^J). On the other hand, each subset J ⊂ S determines a projective, simple G × G-embedding $\mathbb{P}(J)$ of G. The case where J = ∅ is closely related to the wonderful embedding. One obtains a complete list of all subsets J ⊂ S such that $\mathbb{P}(J)$ is a rationally smooth algebraic variety. In such cases we determine the Betti numbers of $\mathbb{P}(J)$ in terms of (W ^J, S ^J). It turns out that $\mathbb{P}(J)$ can be decomposed into a union of “rational” cells. The descent system is used here to help record the dimension of each cell.