# differences between character distributions of supercuspidal representations and others

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?

• I do not know wether or not this overlap what has been said before, however the following fact may answer your question (see Harris notes, Luminy 2007, for instance). Let if $\chi_\pi$ be the character representing $\Theta_\pi$ by Harish-Chandra resularity theorem, i.e. its "density" as a distribution. Then $\pi$ is a discrete series if $\chi_\pi$ does not vanish at some regular elliptic lement of $G$. In that case, it vanishes at all non-elliptic elements. There are also explicit formulas for $\chi_\pi$ for $\pi$ in the principal series. – Desiderius Severus Nov 10 '17 at 9:08
• @DesideriusSeverus, could you give a more precise reference (I mean, for example, where to find the Harris notes)? I believe that it is false that a discrete-series character vanishes at all non-elliptic elements. – LSpice Apr 8 '18 at 2:45
• OK, I found the Harris notes, where the relevant claim is Proposition 1.2.1.3. However, I still believe that it is false. For example, the support of the Steinberg representation of $\mathrm{GL}_2(k)$ contains elements of all tori, split and non-split. (Indeed, I'm pretty sure that the Steinberg character vanishes nowhere on the regular semisimple set, at least for $\mathrm{GL}_2(k)$.) – LSpice Apr 8 '18 at 5:59

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).

• This is morally true far from the identity, but, in the range of the local character expansion, things behave quite differently. – LSpice Apr 8 '18 at 1:38
• @LSpice, it would surely benefit the questioner if you could give specifics... if feasible. I myself do not claim to understand what's happening with those local character expansions. – paul garrett Apr 8 '18 at 2:43
• That is a good idea, but I don't know any results actually characteristic supercuspidal characters purely as scalar functions, only some heuristics and necessary conditions. In a few days I'll see if I can come up with something reasonable to say. (The question being four and a half years old, I guess that there is no rush. :-) ) – LSpice Apr 10 '18 at 16:59

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 mod center 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.

• Deligne's paper works for any group. Casselman's generalisation involves relating the character of even a non-supercuspidal representation to that of its Jacquet restriction to an appropriate parabolic. – LSpice Apr 8 '18 at 1:42
• (Of course by "any group" I mean "any reductive group"; my point was that that generality, which you correctly mention is in Casselman's paper, is already in Deligne's paper.) – LSpice Apr 8 '18 at 6:00
• Finally I think your description of the support is not correct, since an irreducible representation has a central character, so that its support is a union of cosets of the centre; but any element of a compact subgroup of $\mathrm{GL}(N$) has a central translate outside every compact subgroup. What Deligne proves is that every element has compact image in the adjoint quotient; i.e., one might say, is compact modulo centre. – LSpice Dec 19 '18 at 13:22