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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.

In this paper we explicitly determine the Renner monoid ℛ and the cross section lattice Λ of the symplectic algebraic monoid MSpn in terms of the Weyl group and the concept of admissible sets; it turns out that ℛ is a submonoid of ℛn, the Renner monoid of the whole matrix monoid Mn, and that Λ is a sublattice of Λn, the cross section lattice of Mn. Cell decompositions in algebraic geometry are usually obtained by the method of [1]. We give a more direct definition of cells for MSpn in terms of the B × B-orbits, where B is a Borel subgroup of the unit group G of MSpn. Each cell turns out to be the intersection of MSpn with a cell of Mn. We also show how to obtain these cells using a carefully chosen one parameter subgroup.

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