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

Haian He

Published 2018-08-30Version 1

Let $(G,G')$ be a Klein four symmetric pair. If $\pi_K$ is a unitarizable simple $(\mathrm{g},K)$-module, the author shows some necessary conditions when $\pi_K$ is discretely decomposable as a $(\mathfrak{g}',K')$-module. In particular, if $G$ is an exceptional Lie group of Hermitian type, i.e., $G=\mathrm{E}_{6(-14)}$ or $\mathrm{E}_{7(-25)}$, the author classifies all the Klein four symmetric pairs $(G,G')$ such that there exists at least one unitarizable simple $(\mathfrak{g},K)$-module $\pi_K$ that is discretely decomposable as a $(\mathfrak{g}',K')$-module.