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arxiv: 2403.11976 · v3 · submitted 2024-03-18 · 🧮 math.RT · math.NT

On the upper bound of wavefront sets of representations of p-adic groups

Alexander Hazeltine , Baiying Liu , Chi-Heng Lo , Freydoon Shahidi This is my paper

Pith reviewed 2026-05-24 03:15 UTC · model grok-4.3

classification 🧮 math.RT math.NT
keywords wavefront setsp-adic groupsArthur packetsABV packetsanti-discrete seriesreductive groupsadmissible representationsupper bounds
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The pith

A conjecture on upper bounds for wavefront sets of p-adic group representations reduces to anti-discrete series cases and is equivalent to the Jiang conjecture for Arthur packets.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper examines upper bounds on wavefront sets for irreducible admissible representations of connected reductive groups over non-Archimedean local fields of characteristic zero. The authors formulate a new conjecture on these bounds and show that it reduces to the case of anti-discrete series representations, namely those whose Aubert-Zelevinsky duals are discrete series. They establish that the conjecture is equivalent to the Jiang conjecture on upper bounds for representations in local Arthur packets and to an analogous conjecture for local ABV packets. A sympathetic reader would care because these equivalences and the reduction would narrow the verification task to a specific subclass while linking distinct packet structures in the theory of representations over local fields.

Core claim

The formulated conjecture on the upper bound of wavefront sets is equivalent to the Jiang conjecture on the upper bound of wavefront sets of representations in local Arthur packets and also equivalent to an analogous conjecture on the upper bound of wavefront sets of representations in local ABV packets; moreover the conjecture reduces to the case of anti-discrete series representations.

What carries the argument

The new upper-bound conjecture for wavefront sets, which carries the argument through its reduction to anti-discrete series representations via Aubert-Zelevinsky duality.

If this is right

  • If the conjecture holds, then upper bounds on wavefront sets for all irreducible admissible representations follow from verifying them only for anti-discrete series representations.
  • The wavefront set upper bounds for representations in local Arthur packets coincide exactly with those in local ABV packets.
  • Any proof or counterexample for the bounds in the anti-discrete series case immediately settles the general conjecture.
  • The equivalence transfers known or future results on Arthur packets directly to the ABV setting and vice versa.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The reduction might allow computational verification of the bounds first for groups where anti-discrete series representations are well understood, such as classical groups.
  • Matching bounds across packet types could constrain how representations embed into larger L-packets or how their characters behave under endoscopy.
  • If the bounds are sharp in the reduced case, they would limit the possible nilpotent orbits appearing in the wavefront sets of all such representations.

Load-bearing premise

The reduction of the general upper-bound conjecture to the anti-discrete series case is valid, based on the behavior of wavefront sets under Aubert-Zelevinsky duality.

What would settle it

An explicit irreducible admissible representation, especially an anti-discrete series one or one inside a local Arthur packet, whose wavefront set strictly exceeds the conjectured upper bound would disprove the claim.

read the original abstract

In this paper we study the upper bound of wavefront sets of irreducible admissible representations of connected reductive groups defined over non-Archimedean local fields of characteristic zero. We formulate a new conjecture on the upper bound and show that it can be reduced to that of anti-discrete series representations, namely, those whose Aubert-Zelevinsky duals are discrete series. Then, we show that this conjecture is equivalent to the Jiang conjecture on the upper bound of wavefront sets of representations in local Arthur packets and also equivalent to an analogous conjecture on the upper bound of wavefront sets of representations in local ABV packets.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The manuscript formulates a new conjecture on the upper bound of wavefront sets of irreducible admissible representations of connected reductive groups over non-Archimedean local fields of characteristic zero. It proves that this conjecture reduces to the case of anti-discrete series representations (those whose Aubert-Zelevinsky duals are discrete series) and establishes logical equivalence between the new conjecture, the Jiang conjecture for representations in local Arthur packets, and an analogous conjecture for representations in local ABV packets.

Significance. If the reduction via Aubert-Zelevinsky duality and the stated equivalences hold, the paper unifies several wavefront-set conjectures in the representation theory of p-adic groups and reduces the general case to anti-discrete series representations. This logical clarification is a useful organizational contribution that may facilitate future verifications or proofs in the reduced setting; the equivalences themselves constitute the main result.

minor comments (1)
  1. The abstract and introduction would benefit from a brief explicit statement of the precise formulation of the new conjecture (e.g., the precise inclusion or bound asserted for the wavefront set) before discussing its reduction and equivalences.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive report recommending acceptance. The report contains no major comments requiring response.

Circularity Check

0 steps flagged

No circularity; equivalences and reductions are independent logical relations

full rationale

The paper formulates a new conjecture on wavefront set upper bounds, reduces it to the anti-discrete series case using Aubert-Zelevinsky duality (a standard external tool), and proves logical equivalence to the Jiang conjecture and an ABV-packet analogue. These steps are statements of equivalence and reduction between distinct conjectures; none reduce by construction to the paper's own inputs, fitted parameters, or self-citations. No self-definitional loops, renamed empirical patterns, or load-bearing internal citations appear. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract supplies no explicit free parameters or invented entities. The work rests on standard background facts of the field that are not derived inside the paper.

axioms (2)
  • domain assumption Connected reductive groups over non-Archimedean local fields of characteristic zero admit irreducible admissible representations whose wavefront sets are well-defined invariants.
    Stated as the objects of study in the abstract.
  • domain assumption Aubert-Zelevinsky duality maps discrete series to anti-discrete series and preserves relevant representation-theoretic properties needed for the reduction.
    Invoked when the abstract defines anti-discrete series representations via their duals.

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Wavefront sets for genuine representations of $\rm GL$-covers of Kazhdan--Patterson or Savin types

    math.RT 2026-04 unverdicted novelty 5.0

    Wavefront sets for genuine irreps of Kazhdan-Patterson and Savin GL-covers are determined from the degrees of their highest Bernstein-Zelevinsky derivatives, with a Langlands reinterpretation for the former.

Reference graph

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