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 ∩ K)-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 π1 ⊗ π2 of two infinite-dimensional irreducible (g,K)-modules arises as a very special case of our setting. In this case, we prove that π1 ⊗ π2 is discretely decomposable if and only if they are simultaneously highest weight modules.
© Toshiyuki Kobayashi