Towards a tensor-classification of Harish-Chandra modules: The case \rm SL₂(\mathbb R)

2021/04/03 by Januszewski, Fabian
#18D30 (Secondary) #22E45 (Primary) 18D10 #Category Theory (math.CT) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2104.01475

Abstract

We consider the category of Harish-Chandra modules for \rm SL₂(\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 ⊗-submodules of the category of Harish-Chandra modules of \rm SL₂(\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.

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