Let $G$ be a reductive p-adic group, $\pi$ a complex smooth representation of $G$. Then it is known that if $\pi$ is irreducible, then it is admissible.

I need help to find a reference for this fact, and want to know if it is true for real reductive group. Thanks a lot.


1) In these lecture notes:


you have theorem 12, on page 37. He does it for GL_n I think, but it should be similar for other groups...

2) For a real group, I think you should be more specific. There is Soergel's counter-example in the paper "An Irreducible not Admissible Banach Representation of $SL(2, \mathbb{R})$".

  • 1
    $\begingroup$ In addition to the notes of Bernstein, a textbook reference where the result is stated in full generality is Theorem VI.2.2 of Renard: Représentations des groupes réductifs p-adiques. $\endgroup$ – Peter McNamara Nov 10 '18 at 4:38
  • $\begingroup$ @Peter You should post this as an answer. This is a very good reference. $\endgroup$ – Paul Broussous Nov 10 '18 at 5:09

Historically, the result is due to Hervé Jacquet

MR0369624 (51 #5856) Jacquet, Hervé Sur les représentations des groupes réductifs p-adiques. C. R. Acad. Sci. Paris Sér. A-B 280 (1975), Aii, A1271–A1272. 22E50

(From MathSciNet)

"Let F denote a nonarchimedean nondiscrete locally compact field and G a reductive F-group. A representation r or G(F) in a complex vector space V is said to be "smooth'' if the stabilizer of each vector in V is open in G(F); r is said to be "admissible'' if it is smooth and the space of vectors in V fixed by any open compact subgroup of G(F) is finite-dimensional. The author proves that every irreducible smooth representation of G(F) is automatically admissible. The proof, like the result itself, is surprisingly simple and clever. "

  • $\begingroup$ I have looked at this paper and do not believe the proof there is correct. $\endgroup$ – Peter McNamara Nov 29 '18 at 13:25

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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