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

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

2012-02-26

arxiv.org/abs/1202.5743v1

Mathematics

Representation Theory

18 pages

Scientific paper

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,H \cap K)-module. We investigate further if such X can be taken to be a minimal representation, a Zuckerman derived functor module A_q(\lambda), 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 they are simultaneously highest weight modules.

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