The Full Symmetric Toda Flow and Intersections of Bruhat Cells

TL;DR

This paper demonstrates that real Bruhat cells in normal forms of semisimple Lie algebras intersect according to the Bruhat order, using Toda flows rather than geometric analysis, extending complex case results.

Contribution

It establishes the intersection properties of real Bruhat cells in normal forms of semisimple Lie algebras via Toda flows, paralleling complex case results.

Findings

Real Bruhat cells intersect iff their Weyl group elements are comparable in Bruhat order.

The intersection property holds in real normal forms, similar to complex cases.

The approach is based on Toda flows, not geometric analysis.

Abstract

In this short note we show that the Bruhat cells in real normal forms of semisimple Lie algebras enjoy the same property as their complex analogs: for any two elements $w$, $w'$ in the Weyl group $W(\mathfrak g)$, the corresponding real Bruhat cell $X_w$ intersects with the dual Bruhat cell $Y_{w'}$ iff $w\prec w'$ in the Bruhat order on $W(\mathfrak g)$. Here $\mathfrak g$ is a normal real form of a semisimple complex Lie algebra $\mathfrak g_\mathbb C$. Our reasoning is based on the properties of the Toda flows rather than on the analysis of the Weyl group action and geometric considerations.