**[INTRODUCTION]**


Let $G$ be a non-compact simple Lie group, and $G'$ a reductive subgroup of $G$. Suppose that $\pi$ is a non-trivial (hence, infinite dimensional) irreducible unitary representation of $G$ on a Hilbert space, then the restriction $\pi|\_{G'}$ of $\pi$ to $G'$ is **(analytic) discretely decomposable** if $\pi|\_{G'}$ is decomposed as a Hilbert direct sum of irreducible representations of $G'$; namely, it contains no continuous spectrums.


Denote by $\mathfrak{g}$ and $\mathfrak{g}'$ the complexified Lie algebras of $G$ and $G'$ respectively. Take a maximal compact subgroup $K$ of $G$ such that $K':=K\cap G'$ is a maximal compact subgroup of $G'$.


Denote by $\pi\_K$ the underlying (unitarizable simple) $(\mathfrak{g},K)$-module of $\pi$. Then the restriction $\pi\_K|\_{\mathfrak{g}'}$ of $\pi\_K$ to $\mathfrak{g}'$ is **(algebraic) discretely decomposable** if $\pi\_K|\_{\mathfrak{g}'}$ is decomposed as a direct sum of simple $(\mathfrak{g}',K')$-modules.


**[QUESTION]**


It is well-known that if $\pi\_K|\_{\mathfrak{g}'}$ is algebraic discretely decomposable, then $\pi|\_{G'}$ is analytic discretely decomposable. However, if $\pi|\_{G'}$ is analytic discretely decomposable, is $\pi\_K|\_{\mathfrak{g}'}$ algebraic discretely decomposable?


**[THOUGHT]**


Actually, it is known that $\pi\_K|\_{\mathfrak{g}'}$ is algebraic discretely decomposable if and only if there exists a simple simple $(\mathfrak{g}',K')$-module $\tau'\_{K'}$ such that $\mathrm{Hom}\_{\mathfrak{g}',K'}(\tau'\_{K'},\pi\_K|\_{\mathfrak{g}'})\neq\{0\}$. Now $\pi|\_{G'}$ is analytic discretely decomposable, and take a direct summand $\tau'$ which is a subrepresentation. Can $\tau'\_{K'}$ be embeded into $\pi\_K|\_{\mathfrak{g}'}$? I think it not correct because the element in $\tau'\_{K'}$ is not necessarily $K$-finite, right?


I did not find explicit answers for this point in the papers focusing on branching law. I shall be grateful if experts here may share some ideas.



Discrete decomposability of unitary representation
Check the home page of Toshiyuki Kobayashi ([link](https://www.ms.u-tokyo.ac.jp/~toshi/)), download the earlier papers of his, 
over there you fill an answer.


Edit: in a 2017 note of Duflo-Galina-Vargas ([link behind paywall](http://www.heldermann.de/JLT/JLT27/JLT274/jlt27051.htm)), you will find a proof that the answer to your question is yes for discrete series and any pair $(G,G')$. For $A\_q(\lambda)$ and $(G,G')$, you can find the answer in the work of Kobayashi.


