Nilpotent variety of a reductive monoid

In this paper we study the variety M-nil of nilpotent elements of a reductive monoid M. In general this variety has a completely different structure than the variety G(uni) of unipotent elements of the unit group G of M. When M has a unique non-trivial minimal or maximal GxG-orbit, we find a precise description of the irreducible components of M-nil via the combinatorics of the Renner monoid of M and the Weyl group of G. In particular for a semisimple monoid M, we find necessary and sufficient conditions for the variety M-nil to be irreducible.