The H-polynomial of a semisimple monoid

A semisimple monoid M is called quasismooth if M∖{0} has sufficiently mild singularities. We define a cellular decomposition of such monoids using the method of one-parameter subgroups. These cells turn out to be “almost” affine spaces. But they can also be described in terms of the idempotents and B×B-orbits of M. This leads to a number of combinatorial results about the inverse monoid of B×B-orbits of M. In particular, we obtain fundamental information about the H-polynomial of M.