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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 $G'$?

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This question has an open bounty worth +50 reputation from Hebe ending tomorrow.

Looking for an answer drawing from credible and/or official sources.

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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.

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  • $\begingroup$ 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. $\endgroup$ – Hebe May 6 at 15:06

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