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

Abstract

Let [Formula: see text] be a Klein four symmetric pair. The author wants to classify all the Klein four symmetric pairs [Formula: see text] such that there exists at least one nontrivial unitarizable simple [Formula: see text]-module [Formula: see text] that is discretely decomposable as a [Formula: see text]-module. In this paper, three assumptions will be made. First, [Formula: see text] is an exceptional Lie group of Hermitian type, i.e. [Formula: see text] or [Formula: see text]. Second, [Formula: see text] is noncompact. Third, there exists an element [Formula: see text] corresponding to a symmetric pair of anti-holomorphic type such that [Formula: see text] is discretely decomposable as a [Formula: see text]-module.