Conjugacy classes and nilpotent variety of a reductive monoid

We continue in this paper our study of conjugacy classes of a reductive monoid M. The main theorems establish a strong connection with the Bruhat-Renner decomposition of M. We use our results to decompose the variety M-nil of nilpotent elements of M into irreducible components. We also identify a class of nilpotent elements that we call standard and prove that the number of conjugacy classes of standard nilpotent elements is always finite.