\title[Varieties of Nilpotent Elements for Semisimple Lie Algebras]
{Varieties of Nilpotent Elements for Simple Lie Algebras}

\author{UNIVERSITY OF GEORGIA VIGRE ALGEBRA GROUP}

\footnotetext{The members of the UGA VIGRE Algebra Group are
David J. Benson, Phil Bergonio, Brian D. Boe, Leonard Chastkofsky, Bobbe
Cooper, G. Michael Guy, Jo Jang Hyun, Jerome Jungster, Graham
Matthews, Nadia Mazza, Daniel K. Nakano, and Kenyon Platt.}


\begin{abstract} Let $G$ be a simple algebraic group over
$k={\mathbb C}$, or $\overline{{\mathbb F}}_{p}$ where $p$ is good.
Set ${\mathfrak g}=\operatorname{Lie }\ G$. Given $r\in {\mathbb N}$ and a
faithful (restricted) representation $\rho:{\mathfrak g}\rightarrow
\mathfrak {gl}(V)$, one can define a variety of nilpotent elements
${\cal N}_{r,\rho}({\mathfrak g})=\{x\in {\mathfrak g}:\ \rho(x)^{r}=0\}$.
In this paper we determine this variety when $\rho$ is an
irreducible representation of minimal dimension or the adjoint
representation.
\end{abstract}