# Contractions of Representations and Algebraic Families of Harish-Chandra Modules

## TL;DR

This paper studies algebraic families of unitary representations of Lie groups related through contractions, using Harish-Chandra modules and filtrations to understand their structure across continuous and discrete parameters.

## Contribution

It introduces an algebraic framework for families of unitary representations associated with contraction groups, utilizing Harish-Chandra modules and Jantzen filtration.

## Key findings

- Constructed algebraic families of unitary representations
- Analyzed the interplay between discrete and continuous parameters
- Applied Harish-Chandra modules to contraction group representations

## Abstract

We examine from an algebraic point of view some families of unitary group representations that arise in mathematical physics and are associated to contraction families of Lie groups. The contraction families of groups relate different real forms of a reductive group and are continuously parametrized, but the unitary representations are defined over a parameter subspace that includes both discrete and continuous parts. Both finite- and infinite-dimensional representations can occur, even within the same family. We shall study the simplest nontrivial examples, and use the concepts of algebraic families of Harish-Chandra pairs and Harish-Chandra modules, introduced in a previous paper, together with the Jantzen filtration, to construct these families of unitary representations algebraically.