# Exhaustion of restrictions of holomorphic / antiholomorphic representations

Let $$G$$ be a simple Lie group of Hermitian type, and $$G'$$ be a reductive subgroup of $$G$$. Suppose that $$G'$$ is also of Hermitian type and contains the center of the maximal compact subgroup of $$G$$. Thus it is well known that any irreducible (anti)-holomorphic representation (or highest / lowest weight representation) of $$G$$ is discretely decomposable upon restriction to $$G'$$.

Now the question is: if $$\tau$$ is an irreducible (anti)-holomorphic representation of $$G'$$, does there exist an irreducible (anti)-holomorphic representation $$\pi$$ of $$G$$ such that $$\tau$$ is embedded into $$\pi$$ as $$G'$$-representations?

In order words, do the restrictions of irreducible (anti)-holomorphic representations of $$G$$ exhaust all the irreducible (anti)-holomorphic representations of $$G'$$?

I think you're asking about the branching law. Indeed for classical groups over a $$p$$-adic field, the Gan-Gross-Prasad conjecture has two claims, having been proved in several cases, and it should be stated in terms of Langlands correspondence. Gan Wee-tech's papers are good to read, in which there are also examples and observations.
• Thanks for your answer. As you said, the question is about branching law. Actually, I am concerning about the groups over real field. Moreover, I consider the case when $G$ is an exceptional group of Hermitian type though $G'$ may be classical. Is there any reference about it? Thanks again.