Domains of holomorphy for irreducible admissible uniformly bounded representations of simple Lie groups

TL;DR

This paper classifies the domains of holomorphy for irreducible admissible Banach representations of simple Lie groups, providing a complete description for non-Hermitian types and specific cases for Hermitian types.

Contribution

It offers a comprehensive classification of holomorphy domains for irreducible admissible representations, extending previous work to broader Banach space contexts and specific representation types.

Findings

Complete description for non-Hermitian groups' representations.

Determination of holomorphy domains for highest/lowest weight Hermitian cases.

Applicable to uniformly convex and smooth Banach spaces.

Abstract

In this note, we address a question raised by Kr\"otz on the classification of domains of holomorphy of irreducible admissible Banach representations for connected non-compact simple real Lie groups G. When G is not of Hermitian type, we give a complete description of the domains of holomorphy for irreducible admissible uniformly bounded representations on uniformly convex uniformly smooth Banach spaces and, in particular, for all irreducible uniformly bounded Hilbert representations. When the group G is Hermitian, we determine the domains of holomorphy only when the representations considered are highest or lowest weight representations.