Strongly Cuspidal Representations

Updated 16 August 2025

Strongly cuspidal representations are irreducible representations of reductive groups that exhibit enhanced cuspidality by not arising from any parabolic induction.
They are constructed explicitly using techniques such as Deligne–Lusztig theory, compact induction, and type theory, which facilitate their isolation in harmonic analysis.
Their unique formal degree and parameterization play a critical role in classifying representations in Bernstein decompositions and advancing the Langlands correspondence.

A strongly cuspidal representation is an irreducible (typically supercuspidal) representation of a reductive group—over a local field, finite field, or globally—characterized by robust cuspidality properties. Such representations arise as direct analogues or enhancements of the basic notion of cuspidality, typically defined so that the representation is not only absent from the parabolic induction from proper subgroups but also exhibits additional structural or parameter-theoretic constraints. Strongly cuspidal representations play a foundational role in the classification theory for both finite and p-adic groups, in the geometric and spectral structure of the local and global Langlands programs, and in the explicit construction of L-packets, types, and harmonic analysis on algebraic groups.

1. Structural Definition and Existence

A representation π of a reductive group G(F), where F is a non-Archimedean local field (or, in parallel, of a finite group of Lie type or an adele group), is cuspidal if it does not appear as a (sub)quotient of any representation parabolically induced from a proper parabolic subgroup. The “strongly” prefix designates further non-triviality: most precisely, a representation is strongly cuspidal if it is not contained as a subrepresentation anywhere in the induced representations from proper standard cuspidal supports, and (crucially) does not appear as a subquotient of any such parabolic induction—thus, its “matrix coefficients” are compactly supported modulo the center, and the representation is local, isolated, or minimal in the geometric and categorical sense.

A central result is that every reductive group G over a non-Archimedean local field F admits an irreducible, strongly cuspidal complex representation. The explicit construction proceeds via reduction to finite groups of Lie type and the application of Deligne–Lusztig theory: one selects an anisotropic (elliptic) maximal torus T in G and a character θ: T(k) → ℂ× in general position (i.e., with free Weyl group orbit), then considers the virtual Deligne–Lusztig representation R_T^θ. When θ satisfies certain combinatorial non-degeneracy constraints, (–1)^{{σ(G)–σ(T)}} R_T^θ yields an irreducible, strongly cuspidal representation for G(k), and inflation/compact induction yields such for G(F) (Kret, 2012).

The precise “general position” criterion is encoded algebraically: for T₀ a fixed split torus and a Coxeter element w, a class v ∈ X^{(T₀)/(wΦ–1)X^(T₀)} gives rise to a character θ in general position if the orbit of v under ⟨w⟩ is free:

$w^r v ≠ v \quad \text{for all } 1 \leq r < h,$

where h is the Coxeter number. This condition is checked case-by-case for all classical and twisted types.

2. Construction Techniques: Deligne–Lusztig Theory and Types

Deligne–Lusztig theory is the standard framework to construct strongly cuspidal representations for finite groups of Lie type. For a reductive group G/k over a finite field, the procedure is as follows:

Select an elliptic maximal torus T ⊂ G.
Choose a character θ: T(k) → ℂ× whose conjugacy class is “regular” (in general position).
The virtual character R_T^θ is realized as an irreducible representation π_T^θ under favorable conditions. If T is elliptic and θ is regular, π_T^θ is cuspidal (Kret, 2012).

Application to p-adic groups proceeds by inflation of the finite group representations to compact open subgroups, followed by compact induction. In this context, every irreducible strongly cuspidal representation of G(F) can be produced in this way.

For more general groups (e.g., p-adic classical groups and inner forms), constructions proceed via the theory of types—a compactly supported, explicitly constructed pair (J,λ), where J is a compact open subgroup and λ is an irreducible representation, such that the compact induction ind_J^G λ yields an irreducible strongly cuspidal representation. These may be classified by endo-classes or associated semisimple characters and β-extensions.

In the modular and supercuspidal contexts (coefficients of characteristic ℓ ≠ p), the construction of strongly cuspidal (supercuspidal) representations uses elaborations of Bushnell–Kutzko or Yu’s construction, extended to arbitrary coefficient fields (Fintzen, 2019).

3. Parameterization and Uniqueness via Formal Degrees

Strongly cuspidal representations of unipotent or classical types are further characterized by the numerical invariant of their formal degree. For a supercuspidal unipotent representation π of a classical group G over a non-Archimedean local field, the formal degree is a rational function in the residue characteristic q with irreducible factors all powers of q and even cyclotomic polynomials. The crucial uniqueness theorem states that the formal degree of π uniquely determines, up to twisting by weakly unramified characters, its (unramified) Lusztig–Langlands parameter ϕ, via the equality:

$\text{fdeg}(\pi, q) = C_\pi \cdot \gamma(\varphi, q),$

where γ(ϕ, q) is the adjoint gamma factor, and C_π is a representation-dependent constant, often directly computable (Feng et al., 2015).

Thus, the assignment π ↦ ϕ is injective on supercuspidal unipotent representations, and the parameterization—central to the local Langlands correspondence and to harmonic analysis—relies purely on the formal degree.

Spectral transfer morphisms between Hecke algebras arising from these parameters (called “cuspidal STM’s”) are unique up to unramified twist, making the formal degree a complete numerical invariant in this context.

4. Strong Cuspidality on the Galois and Automorphic Sides

From the Galois perspective, strongly cuspidal representations correspond, under the conjectural (and partially established) local Langlands correspondence, to enhanced L-parameters (ϕ, ρ) that are both discrete (do not factor through proper Levi subgroups) and “cuspidal” in the sense of the generalized Springer correspondence (Aubert et al., 2015). This means the associated unipotent element u_ϕ = ϕ(1, $\begin{bmatrix}1 & 1 \ 0 & 1\end{bmatrix}$ ) and the enhancement ρ define a Lusztig-cuspidal pair for the centralizer in the dual group. These are the atomic objects—strongly cuspidal L-parameters—that conjecturally biject with supercuspidal representations of the group.

In the global context (for automorphic representations), strong cuspidality may be associated with representations that are not of CAP (cuspidal associated to a proper parabolic) type, i.e., that do not arise as near-equivalents of Eisenstein series-induced representations. In the function field case, recent advances have shown that test functions isolating the strongly cuspidal spectrum can be constructed purely via Hecke algebra methods, making the separation and identification of the strongly cuspidal automorphic spectrum explicit and effective (Cai et al., 2021).

5. Applications: Harmonic Analysis, Bernstein Decomposition, and Functoriality

The existence and explicit construction of strongly cuspidal representations have wide-ranging applications:

Bernstein decomposition: The category of smooth representations of a p-adic group decomposes into blocks labeled by cuspidal supports. Strongly cuspidal representations form the “atomistic” blocks, and every smooth representation decomposes uniquely as a sum of objects induced from such blocks and subinduced representations from proper parabolics (Meyer, 2015).
Harmonic analysis: Strongly cuspidal representations contribute discretely to the Plancherel formula, enabling the explicit computation of traces and characters for non-inductive spectral terms.
Trace formulas and isolation: Explicit test functions with support in the strongly cuspidal spectrum play a pivotal role in the counting and isolation of automorphic representations, facilitating comparisons in the trace formula relevant to the paper of period integrals and conjectures such as Gan–Gross–Prasad (Cai et al., 2021).
Langlands functoriality and correspondence: The parameterization of strongly cuspidal representations in terms of uniquely determined L-parameters or Galois data (as above) makes the transfer and comparison of automorphic data explicit, as in the case of base change, lifting, and the determining of L-packets.

6. Extensions: Modular, Relative, and Symmetric Varieties

The construction extends to modular representations (coefficients in ℓ ≠ p), where the criterion for strong/supercuspidality can be localized to the inducing type—if the underlying representation A of the compact subgroup J is supercuspidal (as detected via its injective hull in the finite group), then the compact induction ind_J^G A is strongly cuspidal (Henniart et al., 2020).

Further, the theory generalizes to relative and symmetric space settings: for symmetric pairs (G, H), “strongly relatively cuspidal” representations are those irreducible H-distinguished representations whose Jacquet modules with respect to θ-split parabolics vanish. Existence is established under the presence of a maximally θ-split, anisotropic (BG^{––elliptic)} torus in G (Matringe, 10 Jun 2025). These relatively cuspidal representations serve as building blocks in the harmonic analysis and period theory on symmetric spaces.

7. Broader Impact and Perspectives

Strongly cuspidal representations—built via Deligne–Lusztig theory, explicit type-theoretic constructions, and their geometric and combinatorial variants—form the foundational “atomic” spectrum for the representation theory of reductive groups, p-adic groups, and automorphic forms. Their explicit parameterization, uniqueness properties, and role in harmonic analysis make them indispensable in the paper and categorical decomposition of the representation categories, the practical computation of characters and Plancherel measures, and the realization of the local and global Langlands correspondence.

The continued interrogation of their geometric, algebraic, and functional-analytic properties drives much of the current and future progress in the theory of automorphic forms, the Langlands program, and the analytic understanding of the spectra of arithmetic groups.