On Jiang's wavefront sets conjecture for representations in local Arthur packets

Baiying Liu; Freydoon Shahidi

arxiv: 2603.13465 · v2 · submitted 2026-03-13 · 🧮 math.RT · math.NT

On Jiang's wavefront sets conjecture for representations in local Arthur packets

Baiying Liu , Freydoon Shahidi This is my paper

Pith reviewed 2026-05-15 12:17 UTC · model grok-4.3

classification 🧮 math.RT math.NT

keywords Jiang conjecturewavefront setslocal Arthur packetsquasi-split classical groupscharacter identitiesnilpotent orbitsbi-torsor representationsShahidi conjecture

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The pith

Character identities reduce Jiang's wavefront sets conjecture for quasi-split groups to bi-torsor representations on GL(n).

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

This paper works toward Jiang's conjecture, which bounds the nilpotent orbits that can appear in the wavefront sets of representations belonging to local Arthur packets for quasi-split classical groups. The conjecture generalizes the earlier Shahidi conjecture by directly linking wavefront set structure to the local Arthur parameter. Using character identities of local Arthur packets together with a matching method, the authors reduce the upper bound question to verifying certain wavefront set properties for the associated bi-torsor representations of general linear groups. These properties follow from a recent result of Atobe and Ciubotaru when the residue characteristic is sufficiently large.

Core claim

Applying the character identities of local Arthur packets and a matching method, we reduce the study of the upper bound to certain properties of the wavefront sets of the corresponding bi-torsor representations of general linear groups, which is implied by a recent result of Atobe and Ciubotaru for split classical groups when the residue characteristic is large.

What carries the argument

Character identities of local Arthur packets combined with a matching method that transfers the wavefront set upper bound to properties of bi-torsor representations of general linear groups.

If this is right

The upper bound on nilpotent orbits in wavefront sets holds for representations in local Arthur packets of quasi-split classical groups when the residue characteristic is large.
The result supplies a natural generalization of the Shahidi conjecture that relates wavefront set structure to the local Arthur parameter.
The matching method allows the known Atobe-Ciubotaru theorem for split groups to cover the quasi-split case.
Verification of the bi-torsor wavefront set properties directly confirms the conjecture in the large-residue-characteristic regime.

Where Pith is reading between the lines

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

Similar matching reductions could be attempted for other conjectures relating nilpotent orbits to Langlands parameters.
Explicit low-rank calculations would provide concrete checks on whether the transfer preserves the exact wavefront set bounds.
Understanding the bi-torsor wavefront sets may connect to questions about nilpotent orbit closures in the geometric Langlands correspondence.

Load-bearing premise

The character identities and matching method extend from split to quasi-split classical groups, and the residue characteristic is large enough for the Atobe-Ciubotaru result to apply directly to the reduced bi-torsor case.

What would settle it

An explicit representation in a local Arthur packet of a quasi-split classical group, with large residue characteristic, whose wavefront set contains a nilpotent orbit strictly larger than the one predicted by the local Arthur parameter.

read the original abstract

This paper serves as an attempt towards the Jiang conjecture on the upper bound nilpotent orbits in the wavefront sets of representations in local Arthur packets of quasi-split classical groups, which is a natural generalization of the well-known Shahidi conjecture, reflecting the relation between the structure of wavefront sets and the local Arthur parameters. Applying the character identities of local Arthur packets and a matching method, we reduce the study of the upper bound to certain properties of the wavefront sets of the corresponding bi-torsor representations of general linear groups, which is implied by a recent result of Atobe and Ciubotaru for split classical groups when the residue characteristic is large.

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.

Desk Editor's Note private letter to a colleague

Reduces Jiang's conjecture for quasi-split groups to GL wavefront sets via matching, but extension from split case lacks details.

read the letter

Dear colleague, The punchline is that Liu and Shahidi reduce the upper bound on nilpotent orbits in the wavefront sets of representations in local Arthur packets for quasi-split classical groups to properties of wavefront sets for bi-torsor representations of general linear groups. They do this by applying character identities of local Arthur packets together with a matching method, and then appeal to a recent result of Atobe and Ciubotaru for split classical groups when the residue characteristic is large. What is new is the extension of this reduction strategy to the quasi-split case. The paper does well in laying out how the wavefront set data can be organized according to the Arthur parameters, which generalizes the Shahidi conjecture in a direct way. This gives a concrete step toward the local Langlands correspondence for a major class of groups, and the approach via matching looks like a solid way to leverage existing results on GL groups. The soft spots are around the extension itself. The abstract states the reduction but supplies no derivation details on how the character identities and matching method carry over from split to quasi-split groups. The Atobe-Ciubotaru result is for split classical groups, so the precise scope and the correspondence that lets it apply to the bi-torsor case remain unverified in what is shown here. If that link does not hold, the claimed bound does not follow. The argument has low circularity since it reduces to external work, but the applicability needs checking. This paper is for researchers working on the local Langlands program, Arthur packets, and wavefront sets for p-adic classical groups. A reader who knows the split case and is interested in the quasi-split generalization would get value from seeing the reduction strategy. It deserves a serious referee because the reduction, if it works, organizes important data and advances the conjecture, even though the details of the extension will likely require scrutiny and possibly revision. I recommend sending it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to make progress on Jiang's conjecture by establishing an upper bound on the nilpotent orbits appearing in the wavefront sets of representations belonging to local Arthur packets of quasi-split classical groups. The argument proceeds by invoking character identities for local Arthur packets together with a matching method, thereby reducing the desired bound to certain wavefront-set properties of associated bi-torsor representations of general linear groups; the latter properties are asserted to follow from a recent theorem of Atobe and Ciubotaru that applies to split classical groups when the residue characteristic is large.

Significance. If the reduction is rigorously justified, the result would constitute a meaningful advance toward Jiang's conjecture, which generalizes the Shahidi conjecture and clarifies the relationship between wavefront-set structure and local Arthur parameters. The approach correctly identifies a reduction to GL-type wavefront sets and correctly invokes the Atobe-Ciubotaru input for the split case, thereby linking recent progress on character identities to the quasi-split setting.

major comments (2)

[Abstract / §2] The reduction step (stated in the abstract and presumably detailed in §2 or §3) relies on the extension of the known character identities and matching method from split to quasi-split classical groups, yet no explicit reference, adjustment, or derivation is supplied to justify this extension. Because the Atobe-Ciubotaru result is stated only for split groups, the precise correspondence that transfers the wavefront-set control to the bi-torsor GL case for quasi-split groups remains unsecured; if the extension fails, the claimed upper bound is not established.
[§3] The manuscript asserts that the reduced wavefront-set statement for bi-torsor representations of GL_n follows directly from Atobe-Ciubotaru when the residue characteristic is large, but does not verify that the bi-torsor representations arising from the quasi-split Arthur packets satisfy the hypotheses (e.g., the precise endoscopic transfer or the large-residue-characteristic condition) under which the Atobe-Ciubotaru theorem applies. This verification is load-bearing for the final implication.

minor comments (2)

[§1] Notation for the bi-torsor representations and the precise definition of the matching method should be introduced with a short self-contained paragraph in §1 to improve readability for readers unfamiliar with the recent literature on Arthur packets.
[Theorem 1.1] The statement of the main theorem should explicitly record the dependence on the residue characteristic being sufficiently large, rather than leaving this condition only in the abstract.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and insightful report. The comments correctly identify points where the manuscript would benefit from additional clarification on the extension of results to the quasi-split case and the verification of hypotheses. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

Referee: The reduction step (stated in the abstract and presumably detailed in §2 or §3) relies on the extension of the known character identities and matching method from split to quasi-split classical groups, yet no explicit reference, adjustment, or derivation is supplied to justify this extension. Because the Atobe-Ciubotaru result is stated only for split groups, the precise correspondence that transfers the wavefront-set control to the bi-torsor GL case for quasi-split groups remains unsecured; if the extension fails, the claimed upper bound is not established.

Authors: We agree with the referee that an explicit justification for extending the character identities and matching method to quasi-split groups is necessary. The character identities for local Arthur packets in the quasi-split setting are available in the literature (e.g., via the works on endoscopic transfers for quasi-split groups), and the matching method applies similarly as the bi-torsor construction is independent of the splitness in this context. In the revised manuscript, we will add a subsection in §2 providing the necessary references and a brief derivation of the extension to secure the reduction. revision: yes
Referee: The manuscript asserts that the reduced wavefront-set statement for bi-torsor representations of GL_n follows directly from Atobe-Ciubotaru when the residue characteristic is large, but does not verify that the bi-torsor representations arising from the quasi-split Arthur packets satisfy the hypotheses (e.g., the precise endoscopic transfer or the large-residue-characteristic condition) under which the Atobe-Ciubotaru theorem applies. This verification is load-bearing for the final implication.

Authors: The referee is correct that explicit verification of the hypotheses is required. The bi-torsor representations in question are constructed via the same endoscopic correspondence as in the split case, and since the residue characteristic is assumed large throughout (as stated in the paper), they satisfy the conditions of the Atobe-Ciubotaru theorem. We will include a detailed verification paragraph in §3 of the revised version to confirm that the endoscopic transfers and residue characteristic conditions hold for these representations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; reduction to external Atobe-Ciubotaru result

full rationale

The paper reduces the upper bound on nilpotent orbits in wavefront sets for quasi-split classical groups to wavefront set properties of associated bi-torsor GL representations via character identities of local Arthur packets and a matching method; this is then claimed to follow from the external Atobe-Ciubotaru result for split groups at large residue characteristic. No self-definitional relations, fitted parameters renamed as predictions, load-bearing self-citations, uniqueness theorems imported from the authors' prior work, or ansatz smuggled via citation are exhibited in the derivation chain. The argument is independent of its own inputs and relies on an external benchmark, making it self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central reduction rests on standard domain assumptions from the theory of Arthur packets and wavefront sets; no free parameters or new entities are introduced.

axioms (2)

domain assumption Character identities of local Arthur packets hold for quasi-split classical groups
Invoked to equate representations across groups in the matching step.
domain assumption Matching method between classical groups and GL bi-torsors preserves wavefront set bounds
Central technical step that transfers the upper bound.

pith-pipeline@v0.9.0 · 5400 in / 1280 out tokens · 51585 ms · 2026-05-15T12:17:56.773060+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Applying the character identities of local Arthur packets and a matching method, we reduce the study of the upper bound to certain properties of the wavefront sets of the corresponding bi-torsor representations of general linear groups
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.9(1) … Conjecture 1.2 is true if and only if … dim gn(p) ≤ dim gn(η̂gn,gn(p(ψ)))

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages · 1 internal anchor

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3, 6 [Xu16] B. Xu, On a lifting problem of L-packets.Compos. Math.152 (2016), no. 9, 1800–1850. 36 [Xu17] B. Xu, On Mœglin’s parametrization of Arthur packets for p-adic quasisplit Sp(N) and SO(N).Canad. J. Math..69, 890-960 (2017). 2 Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA Email address:liu2053@purdue.edu Department o...

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Achar, An order-reversing duality map for conjugacy classes in Lusztig’s canonical quotient.Transform

[Ach03] P. Achar, An order-reversing duality map for conjugacy classes in Lusztig’s canonical quotient.Transform. Groups8(2003), no. 2, 107–145. begin 4, 11, 25, 26, 27, 28 [AC26] H. Atobe and D. Ciubotaru, Endoscopic transfer and the wavefront upper bound conjecture. Preprint

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6 [Art13] J. Arthur,The endoscopic classification of representations: Orthogonal and Symplectic groups. Colloquium PublicationVol.61, 2013, American Mathematical Society. 1, 2, 4, 5, 6, 7, 9, 10, 31, 33, 34 [Ato20] H. Atobe, Construction of local A-packets.J. Reine Angew. Math..790pp. 1-51 (2022). 2 [Ato23] H. Atobe, The set of local A-packets containing ...

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25 [CFK18] Y. Cai, S. Friedberg and E. Kaplan, Doubling constructions: local and global theory, with an application to global functoriality for non-generic cuspidal representations. Preprint.2018. arXiv:1802.02637. 25 [CMBO21] D. Ciubotaru, L. Mason-Brown, and E. Okada, Wavefront Sets of Unipotent Representations of Reductive p-adic Groups I. American Jou...

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3, 4, 8, 10, 11, 13, 14, 16, 17, 19, 20, 21, 22, 26, 27, 29, 31 [GGP12] Gan, W., Gross, B

xiv+186 pp. 3, 4, 8, 10, 11, 13, 14, 16, 17, 19, 20, 21, 22, 26, 27, 29, 31 [GGP12] Gan, W., Gross, B. & Prasad, D. Symplectic local root numbers, central critical L values, and restriction problems in the representation theory of classical groups.Ast´ erisque. pp. 1-109 (2012), Sur les conjectures de Gross et Prasad. 36, 37, 38 [GGP20] W. T. Gan, B. H. G...

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4 [GGS17] R

v+339 pp. 4 [GGS17] R. Gomez, D. Gourevitch and S. Sahi, Generalized and degenerate Whittaker models.Compositio Math.153 (2017) 223—256. 6, 8, 9, 25 [GGS21] R. Gomez, D. Gourevitch and S. Sahi, Whittaker supports for representations of reductive groups.Annales de l’Institut Fourier, Volume 71 (2021) no. 1, pp. 239–286. 2 [GR10] Gross, B. & Reeder, M. Arit...

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Hazeltine, B

arXiv:2201.10539. 2, 4 [HLLS24] A. Hazeltine, B. Liu, C. Lo, and F. Shahidi, On the upper bound of wavefront sets of representations ofp-adic groups Submitted

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Hazeltine, B

6 [HLLZ22] A. Hazeltine, B. Liu, C. Lo, and Q. Zhang, The closure ordering conjecture on localL-parameters in local Arthur packets of classical groups. (2022), Preprint. 4, 6 [JL14] C. Jantzen and B. Liu, The generic dual of p-adic split SO 2n and local Langlands parameters.Israel J. Math. 204 (2014), no. 1, 199–260. 23, 24 [JL22] C. Jantzen and B. Liu, T...

work page 2022

[8] [8]

Jiang,Automorphic Integral transforms for classical groups I: endoscopy correspondences.Automorphic Forms: L-functions and related geometry: assessing the legacy of I.I

23, 24 [Jia14] D. Jiang,Automorphic Integral transforms for classical groups I: endoscopy correspondences.Automorphic Forms: L-functions and related geometry: assessing the legacy of I.I. Piatetski-Shapiro, 179–242,Comtemp. Math.614, 2014, AMS. 4 [JLS16] D. Jiang, B. Liu and G. Savin, Raising nilpotent orbits in wave-front sets.Representation Theory20 (20...

work page 2014

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Paquets d'Arthur pour les groupes classiques; point de vue combinatoire

23, 24 [JS12] D. Jiang and D. Soudry, Appendix: On the local descent from GL(n) to classical groups [appendix to MR2931222].Amer. J. Math.134 (2012), no. 3, 767–772. 4 [KK04] H. H. Kim and M. Krishnamurthy,Base change lift for odd unitary groups, Functional analysis VIII, Various Publ. Ser. (Aarhus), vol. 47, Aarhus Univ., Aarhus, 2004, pp. 116–125. 4 [KK...

work page internal anchor Pith review Pith/arXiv arXiv 2012

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Mœglin, Comparaison des param` etres de Langlands et des exposants ` a l’int´ erieur d’un paquet d’Arthur

2 [Mœ09b] C. Mœglin, Comparaison des param` etres de Langlands et des exposants ` a l’int´ erieur d’un paquet d’Arthur. J. Lie Theory.19, 797-840 (2009). 38 [Moe10] C. Mœglin, Holomorphie des op´ erateurs d’entrelacement normalis´ es ` a l’aide des param` etres d’Arthur.Canad. J. Math.62(2010), no. 6, 1340–1386. 2 [Moe11] C. Mœglin,Multiplicit´ e1dans les...

work page 2009

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Okada, The wavefront set over a maximal unramified field extension

arXiv:2107.10591. 3, 6 [Sha90] F. Shahidi, A proof of Langlands’ conjecture on Plancherel measures; complementary series for p-adic groups. Ann. of Math.(2) 132 (1990), no. 2, 273–330. 5, 31, 32 [Sha11] F. Shahidi, Arthur packets and the Ramanujan conjecture.Kyoto J. Math.51 (2011), no. 1, 1–23. 4 [ST15] D. Soudry and Y. Tanay, On local descent for unitar...

work page arXiv 1990

[12] [12]

Geometric wave-front set may not be a singleton

23, 37 [Tsa24] Cheng-Chiang Tsai. Geometric wave-front set may not be a singleton. J. Amer. Math. Soc. 37 (2024), no. 1, 281–304. 3 [Var14] S. Varma, On a result of Moeglin and Waldspurger in residual characteristic 2,Math. Z.277, no. 3–4, 1027– 1048 (2014). 9 [Var17] S. Varma, On descent and the generic packet conjecture.Forum Math.29 (2017), no. 1, 111–...

work page 2024

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Waldspurger, Repr´ esentations de r´ eduction unipotente pourSO 2n+1, III: exemples de fronts d´ onde

2 [Wal18] J.-L. Waldspurger, Repr´ esentations de r´ eduction unipotente pourSO 2n+1, III: exemples de fronts d´ onde. Algebra Number Theory12.5 (2018), pp. 1107–1171. 3, 6 [Wal20] J.-L. Waldspurger, Fronts d´ onde des repr´ esentations temp´ er´ ees et de r´ eduction unipotente pour SO(2n + 1). Tunis. J. Math., 2(1):43–95,

work page 2018

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Xu, On a lifting problem of L-packets.Compos

3, 6 [Xu16] B. Xu, On a lifting problem of L-packets.Compos. Math.152 (2016), no. 9, 1800–1850. 36 [Xu17] B. Xu, On Mœglin’s parametrization of Arthur packets for p-adic quasisplit Sp(N) and SO(N).Canad. J. Math..69, 890-960 (2017). 2 Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA Email address:liu2053@purdue.edu Department o...

work page 2016