Abundance of nilpotent orbits in real semisimple Lie algebras

TL;DR

This paper proves that nilpotent orbits are abundant in real semisimple Lie algebras, with hyperbolic elements spanning a key eigenspace, impacting the understanding of fundamental groups in certain geometric spaces.

Contribution

It establishes the abundance of nilpotent orbits and their hyperbolic elements, revealing their spanning properties in the eigenspace related to the Weyl group.

Findings

Hyperbolic elements from nilpotent orbits span the (-1)-eigenspace.

The result links nilpotent orbits to the structure of fundamental groups.

Application to non-Riemannian locally symmetric spaces.

Abstract

We formulate and prove that there are "abundant" in nilpotent orbits 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 span 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.