The following result seems well known.
Let $G$ be a reductive Lie group with a maximal compact subgroup $K$. If $\mu$ is an irreducible unitary representation of $K$, then there exist only finitely many discrete series irreducible unitary representations $\pi$ of $G$ such that $\mu$ appears as a $K$-type in $\pi$.
I do not know where the result was originally showed. I shall be grateful if any expert may offer a reference book or article. Thank you!