Irreducible Supercuspidals of p-adic Groups

• Irreducible supercuspidals are foundational atomic representations in p-adic groups that do not arise from parabolic induction, ensuring maximal cuspidality.
• The topic covers explicit construction methods using Moy–Prasad filtration, Bruhat–Tits buildings, and compact induction to achieve precise classification.
• Implications extend to the local Langlands correspondence, automorphic forms, and harmonic analysis, underscoring their significance in modern representation theory.

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 $N$ (Beuzart-Plessis, 2015, Henniart et al., 2020). 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 (Herzig, 2010).

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 (Herzig, 2010). 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 $G_{x,0}/G_{x,0+}$ (Fintzen, 14 Oct 2025, Aubert, 2023).

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 $\pi = \mathrm{c}$-$\mathrm{Ind}_K^{G(F)}(\bar{\rho} \otimes \kappa)$ (Fintzen, 14 Oct 2025, Aubert, 2023). 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 $\neq p$ (Henniart et al., 2020), with explicit exhaustion and conjugacy classified for quaternionic and classical forms (Skodlerack, 2019).

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 (Vignéras, 2017). The Ollivier–Vignéras criterion asserts that $\pi$ is irreducible admissible supercuspidal iff its pro-$p$ Iwahori invariants form a supersingular $\mathcal H_k$-module (Vignéras, 2017, Herzig et al., 2019).

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 (Herzig, 2010, Herzig et al., 2019).

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 (Vignéras, 2017, Herzig et al., 2019). 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 (Henniart et al., 2020). 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 (Skodlerack, 2019, Henniart et al., 2020).

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 (Henniart et al., 2020).

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 (Fintzen, 14 Oct 2025).

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 (Aubert, 2023). Each compound $L$-packet contains at least one non-singular supercuspidal, with explicit parameters constructed from data on maximal elliptic tori and genericity conditions (Aubert, 2023, Fintzen, 14 Oct 2025). 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 (Blondel et al., 2024).

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 (Vignéras, 2017, Herzig et al., 2019). 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 (Aubert, 2023, Fintzen, 14 Oct 2025, Blondel et al., 2024).

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 (Blondel et al., 2024). These advances confirm conjectural criteria (Moeglin, Shahidi) and further unify the theory across residue characteristics and representation types.