Geometric wave-front set may not be a singleton

Abstract: We show that the geometric wave-front set of a specific type of supercuspidal representations of ramified $p$-adic unitary groups is not a singleton.

• The paper provides an explicit counterexample to the geometric singleton conjecture by constructing U7 epipelagic supercuspidal representations with non-singleton wave-front sets.
• It employs a blend of harmonic analysis, compact induction, and computer-assisted verification to compute local character expansions and nilpotent orbital coefficients.
• The findings reveal that multiple nilpotent orbits can occur in epipelagic settings, prompting a re-evaluation of classical invariants in p-adic representation theory.

Geometric Non-Uniqueness of the Wave-Front Set for Epipelagic Representations of Ramified Unitary Groups

Introduction

This paper investigates the structure of the geometric wave-front set for certain half-integral-depth supercuspidal representations, particularly those of ramified $p$-adic unitary groups. Building on the generalized character expansion of Howe and Harish-Chandra, and the influential results of Mœglin and Waldspurger on the relationship between degenerate Whittaker models and wave-front sets, the work examines a long-standing conjecture: that the geometric wave-front set for irreducible smooth representations is always a singleton, i.e., contained within a single $Ad(G(F^{sep}))$-orbit. Notably, the author provides an explicit counterexample to this conjecture for $G = U_7(E/F)$ in the epipelagic setting, thereby altering the landscape of harmonic analytic invariants in the representation theory of $p$-adic groups.

Background and Technical Setup

For an irreducible smooth $C$-representation $\pi$ of a connected reductive group $G(F)$ over a non-archimedean local field, the Harish-Chandra–Howe local character expansion expresses the character $\Theta_\pi$ as an explicit sum of Fourier transforms of nilpotent orbital integrals in a neighborhood of the identity. The wave-front set of $\pi$ is defined as the subset of maximal nilpotent adjoint orbits $\mathcal{O}$ for which the coefficient $c_{\mathcal{O}}(\pi)$ in this expansion is non-zero; by [MW87], these coefficients coincide with the dimensions of certain degenerate Whittaker models.

For classical groups, wave-front set elements must be special in the Lusztig sense. Previous works for split and unramified groups suggested—conjecturally in general—that the wave-front set is a singleton (up to a geometric conjugacy class). This conjecture is challenged in the context of ramified unitary groups, particularly for the construction undergirding ramified epipelagic representations.

Construction and Main Result

The author works over $F = \mathbb{Q}_3$ with $E/F$ a ramified quadratic extension and considers the ramified unitary group $G = U_7(E/F)$. Supercuspidal representations are constructed via compact induction from characters on first Moy-Prasad filtration subgroups, using an explicit regular semisimple, self-adjoint, and carefully chosen matrix $A$ in $Sym^2(k^7)$, with $k = \mathbb{F}_3$. The main theorem demonstrates:

Any irreducible summand of the compact induction $\operatorname{c-ind}_{G(F)_{1/2}}^{G(F)} \phi_A$ is supercuspidal and has a geometric wave-front set that is not a singleton: specifically, it contains nilpotent orbits of Jordan types $(43)$ and $(511)$.

This is a direct refutation of the aforementioned conjecture for the geometric (maximal nilpotent orbit) wave-front set.

Technical Analysis

Combinatorics of Nilpotent Orbits

For $U_7(E/F)$, the structure of nilpotent orbits is classified by their Jordan types; the relevant orbits are of types $(7)$, $(61)$, $(52)$, $(43)$, and $(511)$. Through a sequence of linear algebra lemmas and computer-aided checks, the author establishes the necessary properties of the chosen $A$ under the adjoint conjugation action, ensuring the inclusion and exclusion of certain orbit types in the support of test functions appearing in the local character expansion.

Harmonic Analysis and Explicit Calculations

By employing harmonic analysis on the Lie algebra and carefully choosing test functions supported in subspaces corresponding to Moy-Prasad lattices, the author computes the coefficients in the local character expansion. The main technical step is to show, for large enough levels, strictly positive values for the character expansion coefficients associated with orbits of type $(43)$ and $(511)$, and zero for orbits of larger Jordan type—thus ensuring the wave-front set is not a singleton.

The proof integrates sophisticated use of the aforementioned character expansion, properties of the compact induction construction, and precise knowledge of orbital integrals, in addition to computer-assisted verification via Magma scripts.

Geometric and Representation-Theoretic Implications

The result relies on fine properties of regular semisimple self-adjoint elements and their stabilizers in the finite orthogonal group, the explicit structure of Moy-Prasad filtrations for ramified groups, and the analytic properties of degenerate Whittaker models. The technical novelty lies in leveraging the structure of the graded versions of the theory (epipelagic representations) and the geometric intricacies of Hessenberg varieties associated to the construction.

Consequences and Further Directions

Counterexamples to the Geometric Singleton Conjecture

This construction shows that, unlike for unramified or depth-zero representations, the geometric wave-front set for certain epipelagic supercuspidal representations can be supported on more than one maximal nilpotent orbit (in terms of inclusion order), thus providing a counterexample to the geometric singleton conjecture. This suggests that the conjecture should be revised or abandoned for wild ramification and epipelagic settings.

Graded Springer Theory and Character Sheaves

The discussion section situates the result in the broader context of graded Springer theory and character sheaves on graded Lie algebras. The author speculates, based on the behavior of the associated Hessenberg varieties, that such non-singleton behavior could be a general phenomenon in the graded setting and for large ranks. The stalks of character sheaves and their counting over finite fields reflect geometric invariants not present in the ungraded case, and these control the analytic behavior of $p$-adic orbital integrals in this context.

Arithmetic Geometric Interplay and Further Generalizations

The connection with arithmetic geometry is instantiated in the explicit role played by point-counts on Hessenberg varieties and stabilizer group schemes. The author indicates, citing [Ts17], that analogous phenomena should exist for higher rank and other types, with the combinatorics of the group action and the structure of epipelagic representations providing the main constraints.

The author embeds the observed representation-theoretic failure of uniqueness into a geometric-analytic framework, suggesting further exploration in the graded/epipelagic regime, the utility of computer verification in constructing concrete examples, and the need for a more refined invariant than the classical notion of geometric wave-front set for wild representations.

Conclusion

This work settles a long-standing ambiguity regarding the uniqueness of geometric wave-front sets for smooth irreducible representations of ramified $p$-adic groups in negative. By constructing explicit epipelagic supercuspidal representations with non-singleton wave-front sets for the ramified unitary group $U_7(E/F)$ when $p=3$, it exposes new phenomena that are invisible in the depth-zero or unramified cases. This points toward a richer geometric structure underlying the representation theory of wild $p$-adic groups, especially in the context of graded Springer theory, character sheaves, and their arithmetic-geometric avatars.

The findings suggest that future studies must reconsider previous conjectures about wave-front sets, accommodate new invariants from arithmetic geometry, and systematically investigate the ramifications of the graded, positive-depth regime in the harmonic analysis of $p$-adic groups.