## Abundance of Nilpotent Orbits in Real Semisimple Lie Algebras

J. Math. Sci. Univ. Tokyo
Vol. 24 (2017), No. 3, Page 399-430.

Okuda, Takayuki
Abundance of Nilpotent Orbits in Real Semisimple Lie Algebras
We formulate and prove that nilpotent orbits are “abundant” in real semisimple Lie algebras, in the following sense. If $S$ denotes the collection of hyperbolic elements corresponding the weighted Dynkin diagrams coming from nilpotent orbits, then $S$ spans the maximally expected space, namely, the $(-1)$-eigenspace of the longest Weyl group element. The result is used to the study of fundamental groups of non-Riemannian locally symmetric spaces.