The H-polynomial of an irreducible representation

Abstract

Let G be a simple algebraic group. Associated with the finite-dimensional rational representation ρ : G → End ( V ) of G there is the monoid M ρ = K ⁎ ρ ( G ) ¯ ⊆ End ( V ) and the projective G × G -embedding P ρ = [ M ρ ∖ { 0 } ] / K ⁎ . One can identify the cases where P ρ is rationally smooth; and in such cases it is desirable to calculate the H-polynomial, H, of P ρ . 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 P ρ . These cells are defined in terms of idempotents, B × B -orbits and other natural quantities obtained from M ρ . Furthermore, H is obtained by recording the dimension of each of these cells in terms of the descent system of M ρ . 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.