Towards a tensor-classification of Harish-Chandra modules: The case ${\rm SL}_2(\mathbb R)$

TL;DR

This paper classifies tensor submodules of Harish-Chandra modules for SL(2,R) using classical principal series results, providing a tensor-based perspective that aligns with classical classifications and aims to extend to broader contexts.

Contribution

It introduces a tensor-classification framework for Harish-Chandra modules for SL(2,R), connecting classical results with new tensor-structural insights.

Findings

Classification of tensor submodules aligns with classical irreducible modules.

Method leverages principal series representations.

Framework potentially generalizes to other reductive pairs.

Abstract

We consider the category of Harish-Chandra modules for ${\rm SL}_2(\mathbb R)$ as a module over the category of finite-dimensional representations of ${\rm SL}(2)$ with respect to the tensor product. In this note we use classical results about principal series to obtain a classification of $\otimes$-submodules of the category of Harish-Chandra modules of ${\rm SL}_2(\mathbb R)$. The resulting classification recovers the classical classification of irreducible Harish-Chandra modules. Our methods are expected to generalize to arbitrary reductive pairs and more general base fields.