# reference help: irreducible implies admissible

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:

http://www.math.tau.ac.il/~bernstei/Unpublished_texts/unpublished_texts/Bernstein93new-harv.lect.from-chic.pdf

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 $SL(2, \mathbb{R})$".

• 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. – Peter McNamara Nov 10 '18 at 4:38
• @Peter You should post this as an answer. This is a very good reference. – 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. "

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