The H-polynomial of a Group Embedding

The Poincaré polynomial of a Weyl group calculates the Betti numbers of the projective homogeneous space G/B, while the h-vector of a simple polytope calculates the Betti numbers of the corresponding rationally smooth toric variety. There is a common generalization of these two extremes called the H-polynomial. It applies to projective, homogeneous spaces, toric varieties and, much more generally, to any algebraic variety X where there is a connected, solvable, algebraic group acting with a finite number of orbits. We illustrate this situation by describing the H-polynomials of certain projective G × G varieties X, where G is a semisimple group and B is a Borel subgroup of G. This description is made possible by finding an appropriate cellular decomposition for X and then describing the cells combinatorially in terms of the underlying monoid of B × B orbits. The most familiar example here is the wonderful compactification of a semisimple group of adjoint type.