  The Poincar\'e 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 $G\times G$-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 $B\times B$-orbits. The
most familiar example here is the wonderful compactification of a semisimple
group of adjoint type.
