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\textsc{Algebra}
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\textsc{Seminar}
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\begin{center}\LARGE
\textit{Rational orbits in the nilpotent variety}
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\begin{center}\Large
George McNinch\par
Tufts University
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\textbf{Abstract:} Let $G$ be a semisimple group over an arbitrary field $K$. Interesting
examples include the special orthogonal group $SO(V, q)$ for a quadratic form q on a
$K$-vector space $V$ and the group $Aut(V)$ for an $8$-dimensional Cayley algebra V . In
these examples, the Lie algebra of $G$ is a Lie subalgebra of $gl(V)$, and the nilpotent
variety — the variety of nilpotent elements of $Lie(G)$ — are just those elements of
$Lie(G)$ which are nilpotent endomorphisms of $V$ .
The group $G$ acts ``by conjugation'' on its Lie algebra, and when $K$ is algebraically
closed it is a basic fact that the number of orbits of $G$ on the nilpotent variety is
finite. When the field $K$ is complete with respect to a discrete valuation and has
finite or algebraically closed residue field, one is often interested in the group $G(K)$
of K-rational points. In the talk, I’ll sketch an argument that the number of orbits
of $G(K)$ on $K$-points of the nilpotent variety remains finite.\vspace{\fill}
\begin{center}\Large
Tuesday, March 20, 2007, 4:00 pm\par
Mathematics and Science Center, W303
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\textsc{Mathematics and Computer Science}\par
\textsc{Emory University}
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