Multiple rational normal forms in Lie theory

TL;DR

This paper investigates a new rational decomposition of elements in reductive complex algebraic groups, involving rational maps and Weyl group elements, expanding understanding of group structure in Lie theory.

Contribution

It introduces a class of rational Weyl group elements that enable specific decompositions of group elements, providing new insights into Lie group structure.

Findings

Defined rational maps N and B for group decomposition

Identified a class of rational Weyl group elements facilitating these decompositions

Analyzed properties and implications of these decompositions in Lie theory

Abstract

We study the decomposition of a generic element $g \in G$ of a connected reductive complex algebraic group $G$ in the form $g = N(g) B(g) \bar{u} N(g)^{-1}$ where $N: G \dashrightarrow \mathcal{N}_-$ and $B : G \dashrightarrow \mathcal{B}_+$ are rational maps onto a unipotent subgroup $\mathcal{N}_-$ and a Borel subgroup $\mathcal{B}_+$ opposite to $\mathcal{N}_-$, and $\bar{u}$ is a representative of a Weyl group element $u$. We introduce a class of rational Weyl group elements that give rise to such decompositions, and study their various properties.