A note on Iwasawa-type decomposition

TL;DR

This paper investigates the Iwasawa-type decomposition of certain subsets of SL(n,C), demonstrating the global dressing action of SU(p,q), the multiplicative nature of admissible elements, and a geometric criterion involving symmetrization.

Contribution

It provides new insights into the structure of admissible elements in AN and their geometric properties under symmetrization, extending understanding of Iwasawa-type decompositions.

Findings

Dressing action of SU(p,q) is globally defined on admissible elements.

Admissible elements form a multiplicative subset of AN.

Symmetrization maps positive cone into itself, providing a geometric criterion.

Abstract

We study the Iwasawa-type decomposition of an open subset of SL(n,C) as SU(p,q)AN. We show that the dressing action of SU(p,q) is globally defined on the space of admissible elements in AN. We also show that the space of admissible elements is a multiplicative subset of AN. We establish a geometric criterion: the symmetrization of an admissible element maps the positive cone in C^n into itself.