Complex groups and root subgroup factorization

TL;DR

This paper studies root subgroup factorization in complex reductive Lie groups, proving the inverse map is rational, providing an algorithm, and exploring explicit formulas for Haar measure and special cases.

Contribution

It establishes the rationality of the inverse root subgroup map, introduces an algorithm involving LDU factorization, and discusses explicit formulas in classical cases.

Findings

Inverse map from root subgroup to triangular coordinates is rational.

An algorithm for the inverse map involving LDU factorization is provided.

Explicit formulas for Haar measure in root subgroup coordinates are derived.

Abstract

Root subgroup factorization is a refinement of triangular (or LDU) factorization. For a complex reductive Lie group, and a choice of reduced factorization of the longest Weyl group element, the forward map from root subgroup coordinates to triangular coordinates is polynomial. We show that the inverse is rational. There is an algorithm for the inverse (involving LDU factorization), and a related explicit formula for Haar measure in root subgroup coordinates. In classical cases there are preferred reduced factorizations of the longest Weyl group elements, and conjecturally in these cases there are closed form expressions for root subgroup coordinates.