A necessary condition for discrete branching laws for Klein four symmetric pairs

Abstract

We show a necessary condition for Klein four symmetric pairs [Formula: see text] satisfying the condition (D.D.); that is, there exists at least one infinite-dimensional simple [Formula: see text]-module that is discretely decomposable as a [Formula: see text]-module. This work is a continuation of [A criterion for discrete branching laws for Klein four symmetric pairs and its application to [Formula: see text], Int. J. Math. 31(6) (2020) 2050049]. Moreover, we define associated Klein four symmetric pairs, and we may use these tools to compute that a class of Klein four symmetric pairs do not satisfy the condition (D.D.); for example, [Formula: see text].