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 G×G-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.