Bounded multiplicity theorems for induction and restriction

We prove a geometric criterion for the bounded multiplicity property of “small” infinite-dimensional representations of real reductive Lie groupsin both induction and restrictions. Applying the criterion to symmetric pairs, we give a full description of the triples $H\subset G\supset G^{\mathrm{\prime }}$ such that any irreducible admissible representations of $G$ with $H$-distinguished vectors have the bounded multiplicity property when restricted to the subgroup $G^{\mathrm{\prime }}$. This article also completes the proof of the general results announced in the previous paper [Adv. Math. 2021, Section 7].