A criterion for discrete branching laws for Klein four symmetric pairs and its application to $E_{6(-14)}$

Abstract

Let $G$ be a noncompact connected simple Lie group, and $(G,G^{\Gamma })$ a Klein four symmetric pair. In this paper, the author shows a necessary condition for the discrete decomposability of unitarizable simple $(\mathfrak{g},K)$-modules for Klein for symmetric pairs. Precisely, if certain conditions hold for $(G,G^{\Gamma })$, there does not exist any unitarizable simple $(\mathfrak{g},K)$-module that is discretely decomposable as a $({\mathfrak{g}}^{\Gamma },K^{\Gamma })$-module. As an application, for $G={\mathrm{E}}_{6(-14)}$, the author obtains a complete classification of Klein four symmetric pairs $(G,G^{\Gamma })$ with $G^{\Gamma }$ noncompact, such that there exists at least one nontrivial unitarizable simple $(\mathfrak{g},K)$-module that is discretely decomposable as a $({\mathfrak{g}}^{\Gamma },K^{\Gamma })$-module and is also discretely decomposable as a $({\mathfrak{g}}^{\sigma },K^{\sigma })$-module for some nonidentity element $\sigma \in \Gamma$.

11 pages, 1 figure. arXiv admin note: text overlap with arXiv:1908.04723

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