Supercuspidal representation

Definition

Let $G=\mathbf G(F)$ for a connected reductive group over a nonarchimedean local field. An irreducible admissible smooth representation $\pi$ of $G$ is supercuspidal if its Jacquet module along every proper parabolic subgroup of $\mathbf G$ is zero. Equivalently, $\pi$ is not a subquotient of parabolic induction from a proper Levi subgroup.

Matrix-coefficient criterion

After fixing a central character, $\pi$ is supercuspidal exactly when its matrix coefficients are compactly supported modulo the center of $G$ . This criterion explains the term “cuspidal”: proper parabolic constant terms vanish.

Position in the classification

Supercuspidal representations are the primitive inputs for the Bernstein decomposition and the p-adic Langlands classification. Parabolic induction from supercuspidal representations of Levi subgroups generates every irreducible smooth representation through subquotients, but a supercuspidal representation itself does not arise from a proper Levi.

Every supercuspidal representation is square-integrable modulo the center, but not every essentially discrete-series representation is supercuspidal.

Parameter-side warning

A “supercuspidal Langlands parameter” is usually a discrete parameter with trivial Deligne monodromy. The assertion that it corresponds precisely to supercuspidal representations is part of the local correspondence and needs hypotheses; it is not the definition above.

References

Colin J. Bushnell and Guy Henniart, The Local Langlands Conjecture for $\mathrm{GL}(2)$ , Springer, 2006, Chapter 1. DOI.
Tasho Kaletha, “Representations of reductive groups over local fields,” §1.2, 2022. arXiv.