On Kostant's partial order on hyperbolic elements

TL;DR

This paper investigates Kostant's partial order on hyperbolic elements in semisimple Lie groups, establishing a converse to a known theorem and exploring its implications in finite-dimensional representations.

Contribution

It proves the converse of a key theorem relating Kostant's partial order to hyperbolic elements, deepening understanding of the structure of semisimple Lie groups.

Findings

Established the converse of [3, Theorem 6.1] for hyperbolic elements

Clarified the relationship between Kostant's partial order and finite-dimensional representations

Enhanced theoretical understanding of semisimple Lie group structure

Abstract

We study Kostant's partial order on the elements of a semisimple Lie group in relations with the finite dimensional representations. In particular, we prove the converse statement of [3, Theorem 6.1] on hyperbolic elements.