Ramified Supercuspidal Components

• Ramified supercuspidal components are irreducible, positive-depth representations induced from compact-mod-center subgroups with intricate ramification structures.
• They are parametrized by tamely or wildly ramified data such as Yu data and admissible characters, which control their internal character theory and role in automorphic forms.
• Their construction involves twisted Levi sequences, Moy–Prasad filtrations, and corresponding Hecke algebras, underpinning key aspects of the local Langlands correspondence and global newforms.

Ramified supercuspidal components are the building blocks—at positive depth and with ramification—of the smooth dual for reductive groups over non-archimedean local fields. These components arise as irreducible supercuspidal representations compactly induced from open compact-mod-center subgroups and are parametrized by tamely or wildly ramified data—such as admissible characters, Yu data, or explicit congruence module representations. Their ramification structure controls not only their internal character theory but also their occurrence in automorphic forms, L-packets, Hecke algebra blocks, and the local Langlands correspondence.

1. Construction via Tame and Ramified Data

The prevailing paradigm in the construction of ramified supercuspidals for $G(F)$ (connected reductive, $F$ non-archimedean) is to specify:

• a twisted Levi sequence $G^0 \subsetneq \cdots \subsetneq G^d=G$, each $G^i$ splitting over a tamely ramified extension,
• a point $x$ in the (enlarged) Bruhat–Tits building $B(G,F)$,
• a sequence of depths $0 < r_0 < \cdots < r_{d-1}$,
• tame generic characters $\phi_i$ of depth $r_i$ on $G^i(F)_{x,r_i : r_i+}$,
• a depth-zero cuspidal datum $\rho$ on $G^0(F)_{x,0}$.

The supercuspidal representation is realized via

$$\pi = \mathrm{c}\text{-Ind}_{K}^{G(F)}(\rho\otimes\kappa),$$

where $K$ is a compact-mod-center subgroup built from the sequence and $\kappa$ is a representation incorporating the Heisenberg–Weil theory from the generic $\phi_i$ (Fintzen, 2019, Fintzen et al., 30 Jan 2025). The tameness assumption ensures well-behaved Moy–Prasad filtration subgroups and, crucially, a finite set of admissible parameterizations for the ramified data.

In the "strongly ramified" case for unitary groups, the data involve a skew maximal simple stratum $[\mathcal{A},n,0,\beta]$ in a Hermitian space, yielding a field $E/F$ with $E/E_0$ a quadratic ramified extension, and a simple character $\theta$, whose lift and level-zero component control both ramification and admissibility (Blondel et al., 2020).

For $GL_2(\mathbb{Q}_p)$, primitive ramified supercuspidals correspond to admissible pairs $(E,\theta_E)$ with $E/\mathbb{Q}_p$ tamely ramified, and are induced from types supported on Iwahori–normalizer subgroups (Loeffler et al., 2010).

2. Ramification Structure and Langlands Parameters

Ramified supercuspidal representations exhibit intricate ramification in their Langlands parameters. The local parameter $\mathcal{L}^{ss}(\pi): W_F \to {}^LG$ is said to be:

• unramified if trivial on inertia $I_F$,
• tamely ramified if trivial on wild inertia $P_F$,
• wildly ramified otherwise.

For groups $G$ not of torus type, if $\pi$ is compactly induced from a sufficiently small open subgroup and pure of weight $0$, then $\mathcal{L}^{ss}(\pi)$ is always (wildly) ramified (Gan et al., 2021). Tame toral parameters, arising from regular characters of tamely ramified maximal tori, correspond to L-parameters factoring through the normalizer and exhibit explicit decomposition under the wild inertia action (Kaletha, 2016, Takase, 2018).

In automorphic settings, the nature of ramified local supercuspidal components determines both the orbit structure and trace formula terms for global newforms with prescribed ramification, with root numbers and ramified extensions governing multiplicities (Knightly et al., 18 Nov 2025, Banerjee et al., 2017).

3. Canonical Models and Geometric Realization

Ramified supercuspidal components can be geometrically attached to explicit mod-$p$ and perverse sheaf data. The theory of commutative character sheaves, as developed for tamely ramified p-adic groups, geometrizes tame quasicharacters and even Weil–Heisenberg constituents on finite-level Greenberg models of parahoric subgroups. The modified category of commutative character sheaves $\mathrm{CCS}(G)$ provides a one-to-one dictionary with $$\mathrm{Hom}(G(\mathbb{F}_q),\overline{\mathbb{Q}}_\ell^\times)$$ modulo vanishing on derived subgroups; this remains valid through ramified covering sequences (Cunningham et al., 2016). The full geometric realization of ramified supercuspidal types employs Lusztig’s character sheaves for finite reductive quotients and Gurevich–Hadani’s geometrization of the Weil representation in the Heisenberg factor.

4. Hecke Algebras, Bernstein Blocks, and Classification

For tame ramified supercuspidals, the attached Hecke algebra $\mathcal{H}(G,\lambda)$ for the type $(K, \lambda)$ is isomorphic to the Hecke algebra of a depth-zero type in an associated twisted Levi subgroup $G^0$, as proved in the "Hecke algebras for tame supercuspidal types" and "Bernstein center of supercuspidal blocks" results (Ohara, 2021, Mishra, 2015). Explicitly,

$$\mathcal{H}(G(F),\lambda) \cong \mathcal{H}(G^0(F), \rho_{-1}),$$

where $\rho_{-1}$ is the depth-zero part of the datum. Consequently, the Bernstein center of a ramified supercuspidal block coincides with that of the corresponding depth-zero block, and the isomorphism forgets higher Moy–Prasad and ramification data.

In the regular (toral) case, the commutative Hecke algebra is isomorphic to the group algebra of the character group of the maximal elliptic torus modulo its maximal compact: $$\mathcal{H}(G(F),\lambda) \simeq \mathbb{C}[S(F)/S(F)_0],$$ encoding in the support the full ramification structure of the supercuspidal character (Ohara, 2021, Kaletha, 2016).

5. Special Cases and Explicit Examples

Several explicit constructions highlight the range of ramified supercuspidal phenomena:

• $SL_n(F)$ and $Sp_{2n}(F)$: For toral (regular) supercuspidals, representations are constructed by compact induction from finite-dimensional representations of $K = SL_n(O_F)$ or $Sp_{2n}(O_F)$, attached to regular elements $\beta$ whose centralizer is an elliptic torus corresponding to a tamely ramified extension $E/F$. Dimension formulas and formal degrees are computed, and the correspondence with L-parameters is established through the local LLC for tori and the Langlands–Shelstad transfer (Takase, 2018, Takase, 2021, Takase, 2021).
• Unitary groups and base change: In the "strongly ramified" scenario, base change for supercuspidal representations between unitary and general linear groups involves careful lifting of simple characters and an explicit sign governed by a quadratic Gauss sum, along with precise analysis of the Hecke algebra's structure constants (Blondel et al., 2020).
• Simple supercuspidals for $GSp_4$: These are constructed at conductor exponent $5$ using affine-generic characters of pro-p Iwahori radicals. Minimal and paramodular new vectors can be explicitly written, and their matrix coefficients and period integrals are evaluated to yield global applications (Pitale et al., 2023).

6. Restriction, Distinction, and L-packet Structure

The restriction of (ramified) supercuspidal representations from a group $G$ to a Levi or derived subgroup is controlled by restricting Yu data, with multiplicities determined by decomposition of the depth-zero representation under the subgroup. In favorable cases (e.g., regular toral data), the restriction is multiplicity-free (Bourgeois, 2020). Furthermore, for groups with Galois involution, the distinction of ramified supercuspidals by a fixed-point subgroup is characterized in terms of invariants attached to the tamely ramified extension and quadratic characters, leading to explicit dichotomies and structure theorems (Sécherre, 2018).

L-packet structure for regular supercuspidals is determined by the stable conjugacy class of the torus and the ramified character data; internal parameterization of the packet translates the ramification structure into cohomological invariants of the parameter (Kaletha, 2016). The formal-degree and root-number conjectures have been checked in full for toral ramified types (Takase, 2021, Takase, 2021).

7. Applications to Automorphic Forms and Global Ramification

Ramified supercuspidal components have precise quantitative impact in the theory of automorphic forms:

• The dimension of spaces of newforms with prescribed (ramified) supercuspidal local components is given in terms of explicit trace formula computations, global root numbers, and class numbers attached to the ramified local extensions, with the Atkin–Lehner operator as a key invariant (Knightly et al., 18 Nov 2025).
• For global period integrals (Novodvorsky, Gan–Gross–Prasad, etc.), explicit formulas for test vectors in ramified supercuspidal components ensure that global decompositions yield the expected local factors, including in higher rank cases (Pitale et al., 2023).

Ramified supercuspidal components also determine the ramification in the local factors of (adjoint) Galois representations attached to motives, with precise local Brauer class and root number criteria following from the local character theory (Banerjee et al., 2017).

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References:

• "On the construction of tame supercuspidal representations" (Fintzen, 2019)
• "Construction of tame supercuspidal representations in arbitrary residue characteristic" (Fintzen et al., 30 Jan 2025)
• "Hecke algebras for tame supercuspidal types" (Ohara, 2021)
• "Bernstein center of supercuspidal blocks" (Mishra, 2015)
• "Commutative character sheaves and geometric types for supercuspidal representations" (Cunningham et al., 2016)
• "Regular supercuspidal representations" (Kaletha, 2016)
• "On supercuspidal representations of $SL_n(F)$ associated with tamely ramified extensions" (Takase, 2018)
• "On certain supercuspidal representations of $SL_n(F)$ associated with tamely ramified extensions: the formal degree conjecture and the root number conjecture" (Takase, 2021)
• "On certain supercuspidal representations of symplectic groups associated with tamely ramified extensions : the formal degree conjecture and the root number conjecture" (Takase, 2021)
• "Base change for ramified unitary groups: the strongly ramified case" (Blondel et al., 2020)
• "Supercuspidal representations of ${\rm GL}_n(F)$ distinguished by a Galois involution" (Sécherre, 2018)
• "Local parameters of supercuspidal representations" (Gan et al., 2021)
• "Counting newforms with prescribed ramified supercuspidal components" (Knightly et al., 18 Nov 2025)
• "Simple supercuspidal representations of $\mathrm{GSp}_4$ and test vectors" (Pitale et al., 2023)
• "Restricting Supercuspidal Representations via a Restriction of Data" (Bourgeois, 2020)
• "Supercuspidal ramifications and traces of adjoint lifts at good primes" (Banerjee et al., 2017)
• "On the computation of local components of a newform" (Loeffler et al., 2010)