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# Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

IMPACT FACTOR 2018: 1.859

CiteScore 2018: 1.14

SCImago Journal Rank (SJR) 2018: 2.554
Source Normalized Impact per Paper (SNIP) 2018: 1.411

Mathematical Citation Quotient (MCQ) 2018: 1.55

Online
ISSN
1435-5345
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Volume 2015, Issue 703

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

Toshiyuki Kobayashi
• Kavli IPMU (WPI) and Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, 153-8914 Tokyo, Japan
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• Other articles by this author:
/ Yoshiki Oshima
Published Online: 2013-07-13 | DOI: https://doi.org/10.1515/crelle-2013-0045

## Abstract

We give a complete classification of reductive symmetric pairs $\left(𝔤,𝔥\right)$ with the following property: there exists at least one infinite-dimensional irreducible (𝔤,K)-module X that is discretely decomposable as an $\left(𝔥,H\cap K\right)$-module. We investigate further if such X can be taken to be a minimal representation, a Zuckerman derived functor module ${A}_{𝔮}\left(\lambda \right)$, or some other unitarizable (𝔤,K)-module. The tensor product ${\pi }_{1}\otimes {\pi }_{2}$ of two infinite-dimensional irreducible (𝔤,K)-modules arises as a very special case of our setting. In this case, we prove that ${\pi }_{1}\otimes {\pi }_{2}$ is discretely decomposable if and only if π1 and π2 are simultaneously highest weight modules.

Published Online: 2013-07-13

Published in Print: 2015-06-01

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

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2015, Issue 703, Pages 201–223, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102,

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