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Classification of symmetric pairs with discretely decomposable restrictions of (𝔤,K)-modules

Toshiyuki Kobayashi and Yoshiki Oshima

Abstract

We give a complete classification of reductive symmetric pairs (𝔤,𝔥) with the following property: there exists at least one infinite-dimensional irreducible (𝔤,K)-module X that is discretely decomposable as an (𝔥,HK)-module. We investigate further if such X can be taken to be a minimal representation, a Zuckerman derived functor module A𝔮(λ), or some other unitarizable (𝔤,K)-module. The tensor product π1π2 of two infinite-dimensional irreducible (𝔤,K)-modules arises as a very special case of our setting. In this case, we prove that π1π2 is discretely decomposable if and only if π1 and π2 are simultaneously highest weight modules.

Funding source: JSPS

Award Identifier / Grant number: Grant-in-Aid for Scientific (A) (25247006)

Funding source: JSPS

Award Identifier / Grant number: Grant-in-Aid for JSPS Fellows

The authors would like to thank the American Institute of Mathematics for supporting the workshop “Branching Problems for Unitary Representations” (July 2011) and the Max Planck Institute for Mathematics for their hospitality, where a part of this project was carried out.

Received: 2012-12-9
Published Online: 2013-7-13
Published in Print: 2015-6-1

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