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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!

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    $\begingroup$ Knapp has something along these lines in (1986, after Corollary 10.37). He points to Harish-Chandra (1953, 1966). $\endgroup$
    – Francois Ziegler
    Commented Oct 4, 2019 at 14:45
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    $\begingroup$ ... and explicitly (1954, Thm 3) where it is called an “immediate consequence of the results proved in (1953)”. $\endgroup$
    – Francois Ziegler
    Commented Oct 4, 2019 at 18:56
  • $\begingroup$ @FrancoisZiegler Thanks a lot! $\endgroup$
    – Hebe
    Commented Oct 5, 2019 at 3:26
  • $\begingroup$ 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.) $\endgroup$
    – Francois Ziegler
    Commented Oct 5, 2019 at 4:51
  • $\begingroup$ It would help if you made more explicit what you mean by "reductive Lie group".(as well as what field you are working over)/ $\endgroup$
    – Jim Humphreys
    Commented Oct 21, 2019 at 1:47

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I believe, the first proof that a K-type occurs at most in finitely many discrete series is in Harish-Chandra

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  • $\begingroup$ Discrete Series II, Acta Math 1?? $\endgroup$
    – jorge vargas
    Commented Oct 20, 2019 at 18:10
  • $\begingroup$ it would be best to add your comment about a more-precise reference to your answer. $\endgroup$
    – paul garrett
    Commented Jul 17, 2020 at 0:49
  • $\begingroup$ Check Lemma 70 in Discrete Series II, Acta math. 166, 1966 $\endgroup$
    – jorge vargas
    Commented Jul 21, 2020 at 23:26

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