On the branching rules for Klein four symmetric pairs

Haian HE

Published 2019-08-13Version 1

Let $G$ be an exceptional simple Lie group of Hermitian type, i.e., $G=\mathrm{E}_{6(-14)}$ or $\mathrm{E}_{7(-25)}$, and $(G,G^\Gamma)$ a Klein four symmetric pair. In this paper, firstly, the author shows that there exists a discrete series representation $\pi$ of $G$ which is $G^\Gamma$-admissible if and only if $(G,G^\Gamma)$ is of holomorphic type. Secondly, the author discusses the discretely decomposable restrictions of $(\mathfrak{g},K)$-modules for the case when $G^\Gamma$ is compact. Finally, the author studies a conjecture by Toshiyuki KOBAYASHI for the associated varieties, and confirms the conjecture for certain pairs.