A note on the Bruhat decomposition of semisimple Lie groups

TL;DR

This paper explores the fixed point structure of split elements acting on flag manifolds in semisimple Lie groups, providing a dynamical proof of the generalized Bruhat decomposition.

Contribution

It introduces a novel dynamical approach to derive the Bruhat decomposition using fixed point analysis of split elements.

Findings

Connected fixed point sets are flag manifolds.

Generalized Bruhat decomposition can be obtained dynamically.

Abstract

Let a split element of a connected semisimple Lie group act on one of its flag manifolds. We prove that each connected set of fixed points of this action is itself a flag manifold. With this we can obtain the generalized Bruhat decomposition of a semisimple Lie group by entirely dynamical arguments.