Algebraic Families of Harish-Chandra Pairs

TL;DR

This paper develops a framework for algebraic families of Harish-Chandra modules to study degenerations and contractions of Lie groups, providing a classification for SL(2, R).

Contribution

It introduces a unified algebraic family framework for Lie group contractions and classifies generically irreducible families for SL(2, R).

Findings

Constructed a family combining real reductive groups and their compact forms.

Classified generically irreducible families of Harish-Chandra modules for SL(2, R).

Unified treatment of contractions within algebraic families.

Abstract

Mathematical physicists have studied degenerations of Lie groups and their representations, which they call contractions. In this paper we study these contractions, and also other families, within the framework of algebraic families of Harish-Chandra modules. We construct a family that incorporates both a real reductive group and its compact form, separate parts of which have been studied individually as contractions. We give a complete classification of generically irreducible families of Harish-Chandra modules in the case of the family associated to SL(2, R).