The H-polynomial of an irreducible representation

Let G be a simple algebraic group. Associated with the finite-dimensional rational representation ρ:G→End(V) of G there is the monoid and the projective G×G-embedding Pρ=[Mρ∖{0}]/K⁎. One can identify the cases where Pρ is rationally smooth; and in such cases it is desirable to calculate the H-polynomial, H, of Pρ. In this paper we consider the situation where ρ is irreducible. We then determine H explicitly in terms of combinatorial invariants of ρ. Indeed, there is a canonical cellular decomposition for Pρ. These cells are defined in terms of idempotents, B×B-orbits and other natural quantities obtained from Mρ. Furthermore, H is obtained by recording the dimension of each of these cells in terms of the descent system of Mρ. As a special case we reacquire the well-known formula for the Poincaré polynomial of a “wonderful embedding” of a simple algebraic group of adjoint type.