Degenerate Moy–Prasad Types Overview

• Degenerate Moy–Prasad types are defined as representation-theoretic data from the Moy–Prasad filtration where nilpotent or unstable functional parameters replace classical semistability.
• They facilitate new decompositions of the Bernstein center by constructing Euler–Poincaré projectors and influencing local harmonic analysis through precise orbital measures.
• Their applications span the local Langlands program, mod-ℓ representations, and geometric representation theory, offering tools for understanding wild ramification and categorified structures.

Degenerate Moy–Prasad types are a class of local representation-theoretic data arising from the Moy–Prasad filtration on reductive groups over non-archimedean local fields. These types generalize the more familiar “nondegenerate” or “epipelagic” types by allowing the underlying character or functional parameters to exhibit nilpotent or nonsemistable behavior. Degenerate Moy–Prasad types play a crucial role in harmonic analysis, the structure theory of the Bernstein center, geometric representation theory, the local Langlands correspondence, and in the paper of mod-$\ell$ representations. Their definition, properties, and applications have been systematically developed across multiple works, enabling new decomposition results and broader understanding of both the representation spaces and the analytic objects acting on them.

1. Definition and Construction of Degenerate Moy–Prasad Types

Degenerate Moy–Prasad types are parametrized by pairs $(x, \chi)$, where $x$ is a point in the Bruhat–Tits building $\mathcal{B}(G)$ of a reductive group $G$ and $\chi$ is a character of the abelian quotient $G_{x,r} / G_{x,r+}$ resulting from the Moy–Prasad filtration at positive depth $r$. Classical (nondegenerate) Moy–Prasad types correspond to functional parameters that are semistable or non-nilpotent under the coadjoint action of the reductive quotient; degenerate types admit nilpotent functionals or characters that do not satisfy semistability.

The formation of these types is particularly nuanced when the underlying invariant forms (e.g., the Killing form) are degenerate—as in the case of $G=\mathrm{SL}_2$ over extensions of $\mathbb{Q}_2$—requiring the use of an alternative pairing, such as the trace pairing, and explicit isomorphisms between multiplicative and additive quotients (e.g., $X \mapsto X-1$) (Kıvran-Swaine, 2011).

In the context of geometric representation theory, a stratum $(x, r, \beta)$ in the loop Lie algebra $\mathfrak{g}((z))$ is said to be degenerate if the representative $\beta_0$ is nilpotent. Degenerate Moy–Prasad types are precisely those strata not fundamental; they fail to detect non-nilpotent behavior and may occur at strictly higher depth than the minimal type (Bremer et al., 2013).

2. Algebraic and Geometric Characterization

Degenerate types are indexed via associate or weak associate classes. For depth-$r$ abelian quotients $G_{F,r}/G_{F,r+}$, a character is “cuspidal” (in the degenerate sense) if it does not extend from any proper parabolic subgroup. Two cuspidal pairs $(F,\chi)$, $(F',\chi')$ are associate if a suitable $g \in G$ aligns $F$ and $F'$ by conjugation and intertwines the characters (Moy et al., 2020).

The notion of weak associativity extends this classification for unrefined minimal $K$-types. Weak associate classes group cosets $Y + \mathfrak{g}_{x,(-r)+}$ into equivalence classes based on nonempty intersection under $G$-conjugacy, producing unions of Bernstein components and the corresponding Bernstein projectors (Kim et al., 2020).

In the mod-$\ell$ setting, degenerate types are characterized by cosets $\phi + \mathfrak{g}^*_{x,>-s}$ containing nilpotent elements. Such types are those local data intersecting nonzero nilpotent orbits, providing key microlocal information for harmonic analysis and character expansions (Tsai, 23 Oct 2025).

3. Functional and Harmonic Analysis Perspective

Bernstein projectors associated to degenerate Moy–Prasad types are constructed as Euler–Poincaré sums over facets of the building, weighted by decorated idempotents $e^{\mathcal{C}}_F$ as dictated by associate classes of cuspidal pairs. The resulting distributions $D^{(\mathcal{C})}$ provide invariant, essentially compact projectors in the Bernstein center and decompose the global depth projector: $$P_{\leq r} = \sum_{\mathcal{C}} D^{(\mathcal{C})}, \qquad D^{(\mathcal{C})} = \sum_{F \subset B_N} (-1)^{\dim F} e^{(\mathcal{C})}_F$$ (Moy et al., 2020). These decompositions also align with direct sum resolutions of representations constructed from spaces of vectors fixed by compact subgroups [Schneider–Stuhler, Bestvina–Savin].

On the Lie algebra side, Bernstein projectors determined by degenerate types are identified via the inverse Fourier transform of characteristic functions supported on unions of good cosets. Push-forward under the logarithm map and harmonic–analytic methods clarify the action of the projector on invariant open subsets of the Lie algebra (Kim et al., 2020).

4. Representation-Theoretic and Category-Theoretic Implications

Degenerate types allow new families of supercuspidal representations. While stable (nondegenerate) types provide irreducibility via compact induction—classically requiring strong stability conditions—degenerate types classified by their root data and commutator relations still yield irreducible, supercuspidal representations, even when stability fails (Gastineau, 2020). Explicit examples for $\mathrm{Sp}_4(\mathbb{Q}_2)$ confirm this possibility.

In categorified settings, the depth filtration corresponding to Moy–Prasad theory gives rise to generation theorems and localization statements for categories of Whittaker sheaves and Kac–Moody modules, with triviality of the Fourier-transformed $D$-module category on degenerate substrata established via devissage arguments and reduction to the unstable locus (Yang, 2021).

Degenerate strata inform the theory of minimal $K$-types for flat $G$-bundles, determining invariants like the slope and enabling the classification of irregular singularities. The formation and refinement of degenerate strata are central to the analysis of wild ramification and moduli (Bremer et al., 2013).

5. Langlands Parameters, Bernstein Center, and Decomposition

Recent work establishes a partial local Langlands correspondence for Moy–Prasad types. Nondegenerate types (with functional parameters not nilpotent) correspond to nontrivial $G^\vee$-conjugacy classes of homomorphisms $\varphi: I_F^r \to G^\vee$ that encode depth-$r$ information through Deligne–Lusztig parameters. Degenerate Moy–Prasad types yield trivial parameters, thus only nondegenerate types contribute actively to the Langlands correspondence (Chen et al., 9 Sep 2025, Bhattacharya et al., 9 Oct 2025).

The category of smooth representations decomposes into products of full subcategories, indexed by these restricted depth-$r$ Langlands parameters. Stable functions on Moy–Prasad quotients and algebra maps into the Bernstein center are used to partition representations into packets indexed by (Deligne–Lusztig/restricted Langlands) parameters, refining the Bernstein decomposition and aligning harmonic-analytic and Galois-theoretic perspectives (Bhattacharya et al., 9 Oct 2025).

6. Character Expansions and Mod-$\ell$ Theory

In the context of mod-$\ell$ representations, the local character expansion leverages the multiplicities of degenerate Moy–Prasad types rather than distribution characters. The expansion: $$v_{(\pi,\psi)}^{>r} = \sum_{O} c_O(\pi, \psi) \cdot v_O^{>r}$$ expresses the occurrence vector of degenerate types in terms of nilpotent orbital measures, with coefficients $c_O(\pi, \psi)$ uniquely determined by triangularity with respect to orbit closures (Tsai, 23 Oct 2025). This approach transfers the analytic difficulties of working with local characters to explicit, computable statistics related to the combinatorics of nilpotent orbits and degeneracy in the Moy–Prasad filtration.

7. Broader Implications and Open Directions

Degenerate Moy–Prasad types provide foundational data for the analysis of representations across various settings: $p$-adic harmonic analysis, geometric representation theory, moduli of $G$-bundles, and the local Langlands program. Their role in decomposing the Bernstein center, constructing projectors, and parametric classification signals deeper connections between pointwise group-theoretic invariants and global Galois-theoretic correspondences. The extension of these methods to rational depths, mod-$\ell$ contexts, and categorical frameworks continues to broaden the impact and scope of Moy–Prasad theory and its degenerate generalizations.

---

Table: Degenerate Moy–Prasad Types and their Associated Structures

Context	Degeneracy Mechanism	Associated Structure
SL₂ over $Q_2$	Degenerate Killing form	Trace pairing, additive duals
Flat $G$-bundles	Nilpotent graded representative $\beta_0$	Slope invariant, wild ramification
Bernstein projector	Cuspidal character not parabolically induced	Euler–Poincaré formula, idempotents
Mod-$\ell$ expansions	Coset intersects nilpotent orbit	Nilpotent orbital integral basis
Langlands parameters	Vanishing under associate class orbit	Trivial parameter in Galois side

Degenerate Moy–Prasad types thus articulate the representation-theoretic and harmonic analytic structures at higher depths or within the nilpotent cone, facilitating intricate realizations of local correspondences, direct sum decompositions, and microlocal analyses in arithmetic and geometric representation theory.