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Let $G$ be a $p$-adic linear reductive group. For an irreducible admissible smooth representation $\pi$ of $G$, there is a distribution $\Theta(\pi)$, called the character distribution, attached to $\pi$. It is given by integration against a locally integrable function which is locally constant on the regular set of $G$.

This distribution in fact determines $\pi$ up to isomorphism.

Since supercuspidal representations are rather special representations of $G$ (e.g. matrix coefficients are compactly supported mod the center), I'm wondering how to determine if a given irreducible representation $\pi$ is supercuspidal in terms of the distribution $\Theta(\pi)$? In others words, which properties of $\Theta$ make supercuspidals different from other smooth representations?

Thanks for any answer.

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2 Answers 2

Since no more-expert person seems to have noticed this question: based on limited experience, and thinking of the archimedean case, in the range between principal series and discrete series, the principal series have their characters smeared out over all of the conjugacy classes of semisimple elements, while at the other end the discrete series characters are supported only on the entirely anisotropic semisimple elements.

The analogue for principal series is easily witnessed in the p-adic case, but I have not personally certified the opposite extreme, discrete series (which includes supercuspidals).

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To make what Paul Garret wrote more precise, let me add that Deligne proved in a CRAS paper that the character of a supercuspidal irreducible representation of GL(N) over a non archimedean local field is supported on the elements contained in some maximal compact subgroup (i.e. fixing some point in the Bruhat-Tits building).

This has been generalized by Casselman to any reductive group in Casselman, W. Characters and Jacquet modules. Math. Ann. 230 (1977), no. 2, 101–105.

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