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Let $\pi$ be an irreducible cuspidal representation of a semi-simple $p$-adic group $G$. It is well-known that the character of $\pi$ is concentrated in the set of compact elements in $G$.

What is the best reference for this result? Also, if there is a really simple proof of this result, I would be glad to learn it!

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  • $\begingroup$ Just want to mention something which probably everybody has already thought of. if we believe in the folklore conjecture that all supercuspidal are compact induction, then the character is supported on (the conjugate of) that compact open. $\endgroup$
    – Cheng-Chiang Tsai
    Sep 30, 2015 at 16:12
  • $\begingroup$ Sure, but my statement is much simpler than that and was probably known already 40 years ago. $\endgroup$
    – Alexander Braverman
    Sep 30, 2015 at 20:36
  • $\begingroup$ As @PaulBroussous mentions, this is originally due to Deligne, but there's also a very nice generalisation due to Casselman (MR). $\endgroup$
    – LSpice
    Mar 8, 2016 at 3:36
  • $\begingroup$ Also, to be fair to @Cheng-ChiangTsai, I believe that also the folklore conjecture—though not its current proof in the tame case by Kim—has been around almost since the dawn of supercuspidals. (Mautner discovered his supercuspidals as compactly induced representations, and, once it became clear that decomposing the Weil representation wasn't the right general approach, it can't have taken long to have supposed that the cInd approach was the only one.) $\endgroup$
    – LSpice
    Mar 8, 2016 at 3:38

1 Answer 1

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This is an old result due to Deligne:

Le support du caractère d'une représentation supercuspidale. C. R. Acad. Sci. Paris Sér. A-B 283 (1976), no. 4, Aii, A155–A157.

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  • $\begingroup$ Thanks you, I knew it was due to Deligne but for some reason I thought it was unpublished. $\endgroup$
    – Alexander Braverman
    Oct 2, 2015 at 14:49

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