Limit orbits in real reductive lie algebras.

Abstract

Let G be a real reductive group, X a semisimple element of the Lie algebra g of G. We define the limit set $$\lim\sb{t\rightarrow0\sp+}t {\bf G}\cdot X\ {:=}\ \{ Y\mid Y=\lim\sb{i\rightarrow\infty}t\sb{i}\cdot Ad(g\sb{i})\cdot X,\ g\sb{i}\ \in {\bf G},\ t\sb{i}\rightarrow0\sp+\}$$ The problem considered in this thesis is to dertermine when this set is the closure of a single G-orbit (necessarily nilpotent). For complex groups this problem, has been solved by W. Borho and H. Kraft (2). For real groups, a conjecture of D. Barbasch and D. Vogan (1) states that $\lim\limits\sb{t\rightarrow\infty}t {\bf G}\cdot X$ is the closure of exactly one real nilpotent orbit where X is an elliptic semisimple element. In this thesis, we do not solve the above problem in general, but we prove a result which provides a reduction to certain Levi subalgebras. This reduction leads to a complete solution in some special cases. In particular, for complex groups we recover a result of W. Borho and H. Kraft.