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.