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

Abstract

We consider the category of Harish-Chandra modules for ${\mathrm{SL}}_2(\mathbb{R})$ as a module over the category of finite-dimensional representations of $\mathrm{SL}(2)$ 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 ${\mathrm{SL}}_2(\mathbb{R})$. 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.