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 $(G,G^Γ)$ 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 $(G,G^Γ)$, there does not exist any unitarizable simple $(\mathfrak{g},K)$-module that is discretely decomposable as a $(\mathfrak{g}^Γ,K^Γ)$-module. As an application, for $G=\mathrm{E}_{6(-14)}$, the author obtains a complete classification of Klein four symmetric pairs $(G,G^Γ)$ with $G^Γ$ noncompact, such that there exists at least one nontrivial unitarizable simple $(\mathfrak{g},K)$-module that is discretely decomposable as a $(\mathfrak{g}^Γ,K^Γ)$-module and is also discretely decomposable as a $(\mathfrak{g}^σ,K^σ)$-module for some nonidentity element $σ\inΓ$.