Supercuspidal Representations in p‑adic Groups

• Supercuspidal representations are irreducible, admissible smooth representations of p‑adic groups that do not arise from parabolic induction.
• Their construction via compact induction, Yu's method, and parametrization through tame elliptic pairs provides precise computational models in representation theory.
• Applications span explicit character formulas, the local Langlands correspondence, and analyses of distinction and period integrals in harmonic analysis.

Supercuspidal representations are irreducible admissible representations of a reductive $p$-adic group $G(F)$ that do not arise as subquotients of any proper parabolically induced representation. They form the atomic objects for harmonic analysis and the local Langlands program for $p$-adic groups, and possess no nontrivial Jacquet modules for any proper parabolic subgroup. Their paper underpins the structure and classification of the entire smooth dual, as well as the analysis of periods, distinction, and L-functions.

1. Foundational Constructions and Parameterization

The construction of supercuspidal representations historically proceeded via compact induction from open subgroups possessing cuspidal data. For $\mathrm{GL}_n(F)$, Howe's original method used characters of elliptic maximal tori embedded into $\mathrm{GL}_n$, extended via finite group theory and Deligne–Lusztig constructions. The introduction of the Bushnell–Kutzko type theory and its systematic exploitation of simple strata and characters allowed a full description for $\mathrm{GL}_n$ and inner forms.

The breakthrough for general reductive groups came with Yu's construction, which for groups $G$ splitting over a tamely ramified extension of $F$, and with residue characteristic large compared to $G$, yields all (tame) supercuspidal representations. A Yu datum $$(G^0 \subset \cdots \subset G^d = G, x, (r_i), \rho, (\phi_i))$$ encodes:

• A sequence of twisted Levi subgroups;
• A point $x$ in the Bruhat–Tits building;
• Depth data $(r_i)$;
• An irreducible depth-zero representation $\rho$ of the compact group $G^0_x$;
• Generic characters $\phi_i$ on each $G^i$.

One obtains a representation

$$\pi = \mathrm{c}\!-\!\mathrm{Ind}_K^{G(F)} (\rho \otimes \kappa)$$

where $K$ is a compact open subgroup built from Moy–Prasad filtration subgroups, and $\kappa$ is constructed as a tensor product of explicit Weil–Heisenberg representations associated to the characters $(\phi_i)$.

Fintzen's results establish that, when $p$ is "well-behaved" (not dividing the order of the absolute Weyl group), every irreducible smooth supercuspidal representation of $G$ arises from such data. For a large class ("regular" supercuspidals), a simpler parametrization by conjugacy classes of pairs $(S, \theta)$—with $S$ a tame elliptic maximal torus and $\theta$ a regular character—suffices (Kaletha, 2016).

2. Character Theory and Explicit Formulas

The Harish–Chandra character of a supercuspidal representation, initially intractable, has become accessible through the formalism developed by Adler, Yu, DeBacker–Spice, and refined by Fintzen, Kaletha, and others (Afgoustidis, 2023, Kaletha, 2016, Chan et al., 2023). For regular supercuspidals, on a domain of "very regular" elements (essentially, topologically semisimple elements whose centralizer is a tame elliptic maximal torus), the character formula specializes to: $$\Theta_{\pi}(s) = \Delta(s)^{-1/2} \cdot \sum_{w \in W(G, S)} \epsilon(w) \cdot \theta(w^{-1} s) \cdot \Delta_{II, S}[a, \chi](s)$$ where:

• $\Delta(s)$ is the Weyl discriminant;
• $\Delta_{II, S}[a, \chi](s)$ is an explicit transfer factor depending on auxiliary $a$-data and $\chi$-data (generalizing the term in Langlands–Shelstad transfer factors);
• The sum is over the Weyl group (modulo $S(F)$), and $\theta$ is the regular character attached to the torus.

When further restricted to "very regular" elements, the complicated orbital integrals and sign corrections collapse, reflecting sharp p-adic analogues of Lusztig's uniqueness theorems for finite groups of Lie type (Chan et al., 2023).

3. Distinction, Symmetry, and Periods

A supercuspidal representation $T$ of $G(F)$ is said to be $H$-distinguished, for a symmetric subgroup $H$, if $\mathrm{Hom}_{H(F)}(T, 1)$ is nonzero. For the particular case of $\mathrm{GL}_n(F)$ with $n$ odd, distinction with respect to an orthogonal subgroup is characterized as follows (Hakim et al., 2011):

• $T$ is $G^\theta$-distinguished if and only if
  • the involution $\theta$ is in the unique split $G$-orbit of orthogonal involutions,
  • and the central character $\omega$ of $T$ satisfies $\omega(-1) = 1$.
  • Furthermore, when distinguished, the space $\mathrm{Hom}_{G^\theta}(T, 1)$ is one-dimensional.

Hakim–Murnaghan's refined multiplicity formula expresses the dimension of the distinguished space as a sum over certain orbits of involutions: $$(O, Y)_G = \sum_{[\theta]} m_{K^0}([\theta]) ([\theta], [Y])_{K^0}$$ with explicit terms computable via Deligne–Lusztig theory and norm indices. In the GL$_n$ case with $n$ odd, all $m_{K^0}([\theta])$ reduce to 1, so distinction is controlled purely by the central character and the conjugacy class of the involution.

On the Langlands side, for regular supercuspidals, distinction is often tied to symmetry of the toral data. For $T(S, \phi)$ attached to a regular pair, $T(S, \phi)$ is $H$-distinguished if and only if the pair $(S, \phi)$ is $(\theta, \epsilon)$-symmetric (Zhang, 2017), i.e.,

• $S$ is $\theta$-stable,
• $\phi|_{S^\theta(F)} = E_S \cdot n_S$ where $E_S$ is a finite-field analogue of the sign character and $n_S$ is a normalization factor.

Such characterization allows the complete determination of all distinguished tame supercuspidals, and the invariants involved are concretely accessible from the inducing torus and character.

4. Connections with the Local Langlands Correspondence

Regular supercuspidal representations have an explicit and canonical parametrization via tame elliptic pairs $(S, \theta)$, which is directly related to their Langlands parameter (Kaletha, 2016). Given a regular character $\theta$ of $S(F)$, one forms the L-parameter via the local Langlands correspondence for tori (relating $\theta$ to a 1-cocycle into the dual torus), then extends via the Langlands–Shelstad transfer (incorporating $\chi$-data for correct transfer factor properties). For each "admissible embedding" of $S$ into $G$, this produces a member of an L-packet. The component group action and the "rectifying" character data determine multiplicities and the structure of the packet.

This parametrization unifies the description of regular supercuspidals with that of real groups (where discrete series are classified by regular characters of compact Cartan subgroups, modulo the Weyl group). The explicit character formulas for $p$-adic regular supercuspidals, when specialized, reproduce Harish–Chandra's formula for real discrete series, highlighting an exact analytic correspondence (Afgoustidis, 2023).

5. Geometric Structures and Methods

The construction and classification of supercuspidal representations rely fundamentally on Bruhat–Tits buildings and Moy–Prasad filtration theory. Each point $x$ in the building determines parahoric and filtration subgroups $G_{x,0}, G_{x,r}$, which model the "depth" and structure of compact open subgroups. This filtration is used to construct the compact induction subgroups in Yu's construction, define minimal $K$-types, and describe the behavior with respect to the centralizer tori.

The representations of finite reductive groups (Deligne–Lusztig theory) are essential as depth-zero building blocks, providing the initial datum (cuspidal representation $\rho$) for the tower in the Yu construction. The explicit behavior of regular characters and Deligne–Lusztig induction is the finite group counterpart of the general phenomenon in $p$-adic groups.

6. Applications and Impact

Supercuspidal representations play a central role in harmonic analysis, trace formula computations, and the understanding of $L$-functions. Distinguished supercuspidals classify periods and appear in relative trace formulas and period integrals (including the Flicker–Kazhdan and Gan–Gross–Prasad cycles). Their explicit character formulas support computations of transfer and stability in endoscopic classification.

The explicit reduction of distinction and multiplicity formulas to concrete invariants (central characters at $-1$, norm groups, symmetry conditions for toral data) allows for direct verification of conjectures such as Prasad's conjecture on dimension of Hom-spaces in quadratic base-change (Wang, 2022), and relations with local period integrals and relative Langlands correspondences.

Recent developments (Fintzen, Kaletha, and collaborators) guarantee the exhaustion and classification of supercuspidal representations within the "tame" case, provide fully explicit character formulas, and establish an explicit local Langlands correspondence for regular $L$-packets (Afgoustidis, 2023, Kaletha, 2016, Chan et al., 2023). These advances have established a robust foundation for further investigations into more general groups, ramified cases, and the paper of types, Hecke algebras, and the overall structure of the smooth dual.

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In summary, the modern theory of supercuspidal representations interlaces explicit algebraic constructions (Yu-data, toral parameters, finite group theory), advanced analytic tools (character formulas, harmonic analysis), and aspects of arithmetic geometry (L-parameters, transfer factors) to yield a detailed, computable, and conceptually unified account of the building blocks of smooth representations of $p$-adic groups. The relative theory of distinction, as developed for tame supercuspidals, not only provides precise local multiplicity criteria but also clarifies their global significance in the Langlands program.