Representation theory of the real Gelfand order and real Harish-Chandra modules for $\mathsf{SL}_2(\mathbbm{R})$

TL;DR

This paper explores the structure of real Harish-Chandra modules for SL(2,R), connecting them to finite-dimensional modules over the real Gelfand order, and classifies key indecomposable representations.

Contribution

It establishes a relationship between real Harish-Chandra modules for SL(2,R) and modules over the real Gelfand order, providing a detailed classification of indecomposables.

Findings

Describes distinguished classes of indecomposable representations.

Relates the category of Harish-Chandra modules to finite-dimensional modules over the real Gelfand order.

Provides a classification framework for modules in the principal block.

Abstract

In this article we study the principal block of the category of real Harish-Chandra modules for the group $\mathsf{SL}_2(\RR)$ and relate it to the category of finite dimensional modules over the so-called real Gelfand order. We describe several distinguished classes of the corresponding indecomposable representations.