---
title: Irreducible Supercuspidals of p-adic Groups
url: https://www.emergentmind.com/topics/irreducible-supercuspidals-of-p-adic-groups
type: topic
---

# Irreducible Supercuspidals of p-adic Groups

Irreducible supercuspidal representations of $p$-adic groups are the foundational building blocks in the smooth representation theory of reductive groups over non-archimedean local fields. These representations exhibit maximal cuspidality: they do not arise as subquotients of any parabolic induction from proper Levi subgroups. In mod-$p$ contexts, the categorically equivalent notion is "supersingular," defined via non-trivial annihilation properties in certain affine Hecke algebras. Their exhaustive construction, explicit classification, and connection to local Langlands parameters underlie much of the arithmetic and harmonic analysis on $p$-adic groups.

## 1. Foundational Definitions and Equivalence Criteria

For any connected reductive group $G$ over a non-archimedean local field $F$ of residue characteristic $p$, an irreducible smooth representation $\pi$ over a coefficient field $C$ is termed **supercuspidal** if $\pi$ does not occur as a subquotient of any parabolic induction $\mathrm{Ind}_P^G(\sigma)$ from a proper parabolic $P=MN$ with Levi $M$ and unipotent radical $N$ [1504.06157, 2010.06462]. In characteristic $p$, supersingularity provides an equivalent criterion: $\pi$ is *supersingular* if all Hecke eigenvalues arise only from the center of the relevant Hecke algebra, precluding inflation from proper Levi subgroups [1005.1713].

For $G=GL_n(F)$, Herzig established that supersingular and supercuspidal are equivalent for irreducible admissible mod-$p$ representations: no parabolic induction or Hecke algebra inflation occurs except from the center [1005.1713]. In general, for split connected reductive groups, this equivalence holds with mild technical hypotheses.

## 2. Explicit Constructions: Moy–Prasad, Bruhat–Tits, and Types

The exhaustive construction of irreducible supercuspidals utilizes the geometry of the Bruhat–Tits building $B(G,F)$ and the Moy–Prasad filtration. Given $x\in B(G,F)$, one defines decreasing filtrations $G_{x,r}$ of compact open subgroups. Depth-zero supercuspidals are classified by pairs $(x, \rho_0)$, with $x$ a vertex and $\rho_0$ an irreducible cuspidal representation of the finite reductive quotient $G_{x,0}/G_{x,0+}$ [2510.12883, 2306.06735].

Yu's construction generalizes to positive-depth cases: it starts with a twisted Levi sequence, depth parameters, generic characters, and a depth-zero building block, assembling these into a compact open subgroup $K$ and a finite-dimensional representation, then forming $\pi = \mathrm{c}$-$\mathrm{Ind}_K^{G(F)}(\bar{\rho} \otimes \kappa)$ [2510.12883, 2306.06735]. Every irreducible supercuspidal of $G(F)$ arises (up to twist) from Yu data under mild tameness and coprimality conditions. Such types also describe supercuspidals in characteristic $\neq p$ [2010.06462], with explicit exhaustion and conjugacy classified for quaternionic and classical forms [1907.02922].

## 3. Hecke Algebras, Supersingularity, and Mod-$p$ Theory

At the algebraic level, the pro-$p$ Iwahori–Hecke algebra $\mathcal H_C(G, \mathcal U)$, for a pro-$p$ Sylow subgroup $\mathcal U$, encodes crucial information. Supersingularity is defined via central Bernstein–Lusztig elements $z_\mu$: a module $M$ is supersingular if $m \cdot z_\mu^n=0$ for sufficiently large $n$ and all $m\in M$ whenever $\mu$ is non-invertible [1712.10142]. The Ollivier–Vignéras criterion asserts that $\pi$ is irreducible admissible supercuspidal iff its pro-$p$ Iwahori invariants form a supersingular $\mathcal H_k$-module [1712.10142, 1905.00053].

In mod-$p$ settings, this strongly influences representation theory. In particular, Herzig's classification for $GL_n$ exhibits supersingular representations as the atomic constituents, showing classification and construction are intimately tied to these properties [1005.1713, 1905.00053].

## 4. Existence, Classification, and Exhaustion Results

The existence of irreducible supercuspidals is unconditional for all connected reductive $p$-adic groups, with explicit exceptions for projective linear groups of division algebras and certain adjoint unitary groups [1712.10142, 1905.00053]. Proofs use discrete cocompact subgroups, global automorphic descent, homological methods on the Bruhat–Tits tree, and Hecke algebra techniques.

Classification proceeds via the compact induction from cuspidal types $(J, \lambda)$, where $J$ is a carefully chosen open subgroup and $\lambda$ an irreducible representation satisfying an intertwining–implies–conjugacy property [2010.06462]. For many classes (Bushnell–Kutzko, Stevens, Deligne–Lusztig), these types are explicitly described; exhaustion and unicity up to conjugacy hold for broad families, including symmetric, unitary, and symplectic groups [1907.02922, 2010.06462].

A practical supercuspidality criterion is: $\mathrm{c}$-$\mathrm{Ind}_J^G \lambda$ is supercuspidal iff the finite reductive quotient representation $\bar{\lambda} \in \mathrm{Irr}_C(J^0/J^1)$ is itself supercuspidal [2010.06462].

## 5. Character and Parameter Theory: Local Langlands and Internal Structure

Supercuspidal representations admit Harish–Chandra character formulas, often showing parallels to real discrete series. For $p$-adic regular supercuspidals, Kaletha's torus-formula expresses the character as a Weyl-type sum over the normalizers of elliptic maximal tori and their generic characters [2510.12883].

Under the local Langlands correspondence (LLC), irreducible supercuspidals correspond (in tame, coprime settings) to enhanced $L$-parameters with semisimple cuspidal support specified by the Springer correspondence [2306.06735]. Each compound $L$-packet contains at least one non-singular supercuspidal, with explicit parameters constructed from data on maximal elliptic tori and genericity conditions [2306.06735, 2510.12883]. For symplectic groups, recent results use explicit Fourier expansions and $\gamma$-factor criteria to fully determine Langlands parameters for simple supercuspidals and their exact points of reducibility [2406.15767].

## 6. Consequences, Applications, and Recent Advances

The existence and classification of irreducible supercuspidals underpin the structure of the Bernstein decomposition, automorphic form constructions, and the mod-$p$ and $l$-modular Langlands correspondences. Their role as atomic modules is fundamental for harmonic analysis, cohomology of Shimura varieties, and the explicit realization of automorphic types [1712.10142, 1905.00053]. Exhaustive parameterizations now exist for broad classes of classical groups, and links to arithmetic are realized via LLC, with $L$-packets and character formulas determined by geometric invariants and Springer-theoretic data [2306.06735, 2510.12883, 2406.15767].

Recent work extends the uniform construction and classification to positive characteristic fields, determining explicit LLC-parameters for simple supercuspidals, mapping their functorial transfers, and exactly characterizing reducibility points via local $\gamma$-factors [2406.15767]. These advances confirm conjectural criteria (Moeglin, Shahidi) and further unify the theory across residue characteristics and representation types.

Source: https://www.emergentmind.com/topics/irreducible-supercuspidals-of-p-adic-groups