Non-compact groups of inner type and factorization

TL;DR

This paper extends root subgroup and Birkhoff factorizations from compact to noncompact simple Lie groups of inner type, providing new characterizations and measure properties in this broader context.

Contribution

It introduces parallel characterizations of Birkhoff components and root subgroup coordinates for noncompact inner type Lie groups, generalizing known compact group results.

Findings

Birkhoff components characterized for noncompact groups

Root subgroup coordinates constructed for these components

Haar measure restricted to top Birkhoff component is a product measure

Abstract

We investigate Birkhoff (or triangular) factorization and (what we propose to call) root subgroup factorization for elements of a noncompact simple Lie group $G_0$ of inner type. For compact groups root subgroup factorization is related to Bott-Samelson desingularization, and many striking applications have been discovered by Lu (\cite{Lu}). In this paper, in the inner noncompact case, we obtain parallel characterizations of the Birkhoff components of $G_0$ and an analogous construction of root subgroup coordinates for the Birkhoff components. As in the compact case, we show that the restriction of Haar measure to the top Birkhoff component is a product measure in root subgroup coordinates.