# K-type in discrete series representation

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!

• Knapp has something along these lines in (1986, after Corollary 10.37). He points to Harish-Chandra (1953, 1966). – Francois Ziegler Oct 4 '19 at 14:45
• ... and explicitly (1954, Thm 3) where it is called an “immediate consequence of the results proved in (1953)”. – Francois Ziegler Oct 4 '19 at 18:56
• @FrancoisZiegler Thanks a lot! – Hebe Oct 5 '19 at 3:26
• I didn’t make this an answer, because H-C’s context is slightly different: fixed infinitesimal character. If you can add how it implies what you wanted, don’t hesitate to post that as self-answer. (I haven’t thought about it.) – Francois Ziegler Oct 5 '19 at 4:51
• It would help if you made more explicit what you mean by "reductive Lie group".(as well as what field you are working over)/ – Jim Humphreys Oct 21 '19 at 1:47