Discretely decomposable restrictions of $(\mathfrak{g},K)$-modules for Klein four symmetric pairs of exceptional Lie groups of Hermitian type

Abstract

Let $(G,G^{\Gamma })$ be a Klein four symmetric pair. The author wants to classify all the Klein four symmetric pairs $(G,G^{\Gamma })$ such that there exists at least one nontrivial unitarizable simple $(\mathfrak{g},K)$-module ${\pi }_K$ that is discretely decomposable as a $({\mathfrak{g}}^{\Gamma },K^{\Gamma })$-module. In this article, three assumptions will be made. Firstly, $G$ is an exceptional Lie group of Hermitian type, i.e., $G={\mathrm{E}}_{6(-14)}$ or ${\mathrm{E}}_{7(-25)}$. Secondly, $G^{\Gamma }$ is noncompact. Thirdly, there exists an element $\sigma \in \Gamma$ corresponding to a symmetric pair of anti-holomorphic type such that ${\pi }_K$ is discretely decomposable as a $({\mathfrak{g}}^{\sigma },K^{\sigma })$-module.

8 pages, 1 figure