QUESTION
Let $G$ be a simple Lie group with equal rank; namely, the rank of $G$ equals the rank of its maximal compact subgroup. Suppose that $G'$ is a reductive subgroup of $G$ with equal rank. If $\pi$ is a discrete series representation of $G$, is its restriction $\pi|_{G'}$ a discrete series representation of $G'$? In particular, what if $(G,G')$ is a symmetric pair?
IDEA
There are many papers studying the decrete decomposability of the restriction of discrete series (or $A_\mathfrak{q}(\lambda)$ as $(\mathfrak{g},K)$-module) for symmetric pairs. However, there seems no explicit statement for the question above. Actually, the question is whether $L^2(G)\subseteq L^2(G')$ holds. If $M'$ is a closed submanifold of $M$, a smooth function $f\in L^2(M)$ does not imply $f\in L^2(M')$ in general. However, I am not sure for the case for reductive Lie groups, especially for symmetric pairs.
