Analytic representation theory of Lie groups: General theory and analytic globalizations of Harish--Chandra modules

TL;DR

This paper develops a comprehensive framework for analytic representations of real Lie groups, introduces temperedness, and proves the existence and uniqueness of tempered analytic globalizations for Harish-Chandra modules.

Contribution

It introduces a general theory of analytic representations, defines temperedness, and establishes the unique tempered analytic globalization for Harish-Chandra modules.

Findings

Temperness indicates an action of the algebra A(G) of rapidly decaying analytic functions.

Every Harish-Chandra module admits a unique tempered analytic globalization.

The tempered globalization embeds as the space of analytic vectors in all Banach globalizations.

Abstract

In this article a general framework for studying analytic representations of a real Lie group G is introduced. Fundamental topological properties of the representations are analyzed. A notion of temperedness for analytic representations is introduced, which indicates the existence of an action of a certain natural algebra A(G) of analytic functions of rapid decay. For reductive groups every Harish-Chandra module V is shown to admit a unique tempered analytic globalization, which is generated by V and A(G) and which embeds as the space of analytic vectors in all Banach globalizations of V.