7
$\begingroup$

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!

$\endgroup$
8
  • 1
    $\begingroup$ Knapp has something along these lines in (1986, after Corollary 10.37). He points to Harish-Chandra (1953, 1966). $\endgroup$ Oct 4, 2019 at 14:45
  • 1
    $\begingroup$ ... and explicitly (1954, Thm 3) where it is called an “immediate consequence of the results proved in (1953)”. $\endgroup$ Oct 4, 2019 at 18:56
  • $\begingroup$ @FrancoisZiegler Thanks a lot! $\endgroup$
    – Hebe
    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.) $\endgroup$ 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)/ $\endgroup$ Oct 21, 2019 at 1:47

1 Answer 1

1
$\begingroup$

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

$\endgroup$
3
  • $\begingroup$ Discrete Series II, Acta Math 1?? $\endgroup$ Oct 20, 2019 at 18:10
  • $\begingroup$ it would be best to add your comment about a more-precise reference to your answer. $\endgroup$ Jul 17, 2020 at 0:49
  • $\begingroup$ Check Lemma 70 in Discrete Series II, Acta math. 166, 1966 $\endgroup$ Jul 21, 2020 at 23:26

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.