Arithmetic Wavefront Set and Microlocal Structure of Harish-Chandra Character

Dihua Jiang; Dongwen Liu; Jia-Jun Ma; Lei Zhang; Zhikang Luo

arxiv: 2606.05735 · v1 · pith:TOQXZTCMnew · submitted 2026-06-04 · 🧮 math.RT

Arithmetic Wavefront Set and Microlocal Structure of Harish-Chandra Character

Dihua Jiang , Dongwen Liu , Zhikang Luo , Jia-Jun Ma , Lei Zhang This is my paper

Pith reviewed 2026-06-27 23:24 UTC · model grok-4.3

classification 🧮 math.RT

keywords wavefront setsHarish-Chandra characterslocal L-parametersclassical groupsCasselman-Wallach representationsmicrolocal structurenilpotent orbitsdegenerate Whittaker models

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The pith

For representations with generic local L-parameters, the wavefront set from the Harish-Chandra character equals the arithmetic wavefront set defined by the enhanced L-parameter.

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

The paper proves reciprocity between wavefront sets for irreducible admissible representations of classical groups over local fields of characteristic zero whenever the representation has a generic local L-parameter. Over the reals, for Casselman-Wallach representations, this equality shows that the microlocal structure of the distribution character is fixed entirely by the arithmetic data in the enhanced L-parameter. The result confirms the Wavefront Set Conjecture and its refinement, and yields a sharpened form of Vogan's maximal-orbit principle. The algebraic wavefront set defined by degenerate Whittaker models enters through a composition law that holds uniformly across all such fields.

Core claim

For an irreducible Casselman-Wallach representation π with a generic local L-parameter, the arithmetic wavefront set WF_ari(π) defined by the enhanced local L-parameter coincides with the wavefront set WF_tr(π) defined by the Harish-Chandra character Θ_π; hence the microlocal structure of Θ_π is completely determined by the arithmetic information carried by the enhanced local L-parameter. The same reciprocity holds for irreducible admissible representations over any local field of characteristic zero with generic L-parameter, and the algebraic wavefront set is related to the others by the composition law.

What carries the argument

The enhanced local L-parameter of the representation, which defines the arithmetic wavefront set and determines the singularities of the distribution character.

If this is right

The Wavefront Set Conjecture and its refinement hold for all such representations over archimedean fields.
A refinement of Vogan's maximal-orbit principle follows as a consequence.
The algebraic wavefront set defined by degenerate Whittaker models is related to the arithmetic and trace wavefront sets by the composition law over every local field of characteristic zero.
The microlocal structure of the character is fixed by arithmetic data alone when the L-parameter is generic.

Where Pith is reading between the lines

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

The same matching might hold for some non-generic parameters if the generic hypothesis can be relaxed by other means.
Explicit character computations for classical groups could be reduced to calculations on the enhanced L-parameter side.
The composition law may supply a uniform way to compare wavefront sets across different realizations of the same representation.

Load-bearing premise

The representation possesses a generic local L-parameter, and over the reals the argument further relies on earlier results plus an auxiliary conjecture.

What would settle it

An explicit irreducible representation with generic local L-parameter whose Harish-Chandra character has a wavefront set different from the nilpotent orbit predicted by its enhanced L-parameter.

read the original abstract

In this paper, we establish in Theorem 1.2 the reciprocity of wavefront sets for irreducible admissible representations $\pi$ of classical groups $G$ over any local field $F$ of characteristic zero if $\pi$ has a generic local $L$-parameter. Over archimedean local fields, based on the progress made in our previous work (arXiv:2207.04700), we prove in Theorem 1.5 that for an irreducible Casselman--Wallach representation $\pi$ with a generic local $L$-parameter, the Wavefront Set Conjecture (arXiv:2207.04700, Conjecture 1.2) and its refinement (Conjecture 1.1) hold for the arithmetic wavefront set ${\mathrm{WF}}_{\mathrm{ari}}(\pi)$ as defined by the associated enhanced local $L$-parameter of $\pi$ and the wavefront set ${\mathrm{WF}}_{\mathrm{tr}}(\pi)$ defined by the Harish--Chandra distribution character $\Theta_\pi$ of $\pi$. Hence the microlocal structure of $\Theta_\pi$ is completely determined by the arithmetic information carried by the enhanced local $L$-parameter of $\pi$. The relations with the algebraic wavefront set ${\mathrm{WF}}_{\mathrm{wm}}(\pi)$ defined by the degenerate Whittaker models are extensively discussed by means of the composition law (Theorem 3.4) over all local fields of characteristic zero. Under Conjecture 1.3, the Wavefront Set Conjecture is fully established over archimedean local fields. As a consequence, we prove a refinement of Vogan's maximal-orbit principle (Theorem 1.7).

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

This paper proves reciprocity and the wavefront set conjecture for generic L-parameters, extending the authors' prior work while leaving the full archimedean case conditional on one extra conjecture.

read the letter

The main advance is Theorem 1.2, which establishes reciprocity between arithmetic and trace wavefront sets for any irreducible admissible representation of a classical group over a local field of characteristic zero, provided the local L-parameter is generic. Over the reals they also prove Theorem 1.5, showing that the wavefront set conjecture and its refinement hold under the same generic hypothesis, so the microlocal structure of the Harish-Chandra character is fixed by the enhanced L-parameter. The composition law in Theorem 3.4 then relates the arithmetic, trace, and algebraic wavefront sets across all such fields, and this yields a refinement of Vogan's maximal-orbit principle in Theorem 1.7.

The work is a direct continuation of their arXiv:2207.04700 paper. The new pieces are the reciprocity statement and the composition law; both appear cleanly stated and scoped to the generic case. The archimedean proofs rest on the earlier results plus the standing Conjecture 1.3, which the authors isolate explicitly.

The dependence on the previous paper is the clearest soft spot, but it is acknowledged rather than hidden. The generic-L-parameter hypothesis is a standard restriction in this area and the claims are not overstated once that is taken into account. No internal contradictions or unsupported steps show up in the abstract or the stress-test summary.

This is for specialists working on microlocal properties of characters and nilpotent orbits in the Langlands program for classical groups. A reader already following the wavefront-set literature will get concrete new relations and a partial resolution of the conjecture. I would send it to peer review; the results are scoped and the progress on the conjecture is real enough to merit referee attention.

Referee Report

2 major / 2 minor

Summary. The paper claims to establish the reciprocity of wavefront sets for irreducible admissible representations π of classical groups G over local fields F of characteristic zero with generic local L-parameter, equating WF_ari(π) from the enhanced L-parameter and WF_tr(π) from the Harish-Chandra character Θ_π (Theorem 1.2). Over archimedean fields, it proves the Wavefront Set Conjecture and refinement for Casselman-Wallach representations with generic L-parameter, determining the microlocal structure of Θ_π by the L-parameter (Theorem 1.5), relying on prior work arXiv:2207.04700 and Conjecture 1.3. It discusses relations to the algebraic wavefront set WF_wm(π) via the composition law (Theorem 3.4) and proves a refinement of Vogan's maximal-orbit principle (Theorem 1.7).

Significance. If the results hold, this work significantly advances the understanding of wavefront sets by linking arithmetic information from enhanced local L-parameters to the microlocal properties of distribution characters in the theory of representations of classical groups. The explicit treatment of the composition law relating different wavefront set definitions across all local fields of characteristic zero is a notable strength, as is the clear isolation of the remaining Conjecture 1.3 for the full archimedean case. The paper provides credit to the dependence on prior results and states hypotheses explicitly.

major comments (2)

[§1, Theorem 1.5] §1, Theorem 1.5: The claim that the microlocal structure of Θ_π is 'completely determined' by the enhanced local L-parameter holds only conditionally on the results of arXiv:2207.04700 and the validity of Conjecture 1.3; these are load-bearing for the archimedean case and are not established within the manuscript.
[Theorem 1.2] Theorem 1.2: The reciprocity is asserted uniformly over any local field of characteristic zero under the generic L-parameter hypothesis, but the text indicates the archimedean proof builds directly on the prior arXiv:2207.04700 while the non-archimedean case appears independent; a explicit case distinction or uniform argument reference is needed to confirm the claim is fully supported.

minor comments (2)

[§3, Theorem 3.4] §3, Theorem 3.4: The composition law is used to relate WF_ari, WF_tr and WF_wm; recalling the precise definitions of these three sets at the start of the section would improve self-contained readability.
The abstract states that relations with WF_wm(π) are 'extensively discussed'; a single summarizing corollary or table collecting the equalities/inequalities under the generic L-parameter hypothesis would aid navigation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the precise comments on the manuscript. We address the two major comments point by point below. Both points concern clarity of dependence on prior results and we agree that modest revisions will strengthen the exposition.

read point-by-point responses

Referee: [§1, Theorem 1.5] §1, Theorem 1.5: The claim that the microlocal structure of Θ_π is 'completely determined' by the enhanced local L-parameter holds only conditionally on the results of arXiv:2207.04700 and the validity of Conjecture 1.3; these are load-bearing for the archimedean case and are not established within the manuscript.

Authors: We agree that the determination of the microlocal structure in the archimedean case is conditional on the results of arXiv:2207.04700 together with the validity of Conjecture 1.3. The manuscript already records this dependence in the abstract, the statement of Theorem 1.5, and the surrounding discussion. To make the conditional character fully explicit, we will revise the wording of Theorem 1.5 and add a short clarifying sentence in §1 emphasizing that the claim relies on those external inputs. revision: yes
Referee: [Theorem 1.2] Theorem 1.2: The reciprocity is asserted uniformly over any local field of characteristic zero under the generic L-parameter hypothesis, but the text indicates the archimedean proof builds directly on the prior arXiv:2207.04700 while the non-archimedean case appears independent; a explicit case distinction or uniform argument reference is needed to confirm the claim is fully supported.

Authors: The reciprocity asserted in Theorem 1.2 is indeed uniform under the generic L-parameter hypothesis, with the non-archimedean direction proved independently in the present paper and the archimedean direction relying on arXiv:2207.04700. We will insert an explicit case distinction (or a brief reference to the respective foundations) immediately after the statement of Theorem 1.2 to remove any ambiguity about the logical structure of the argument. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained under stated hypotheses

full rationale

The paper conditions Theorems 1.2 and 1.5 on the explicit hypothesis that π possesses a generic local L-parameter. The archimedean case builds on prior independent results from arXiv:2207.04700 without reducing the new reciprocity or microlocal claims to self-definitional equivalences, fitted inputs renamed as predictions, or load-bearing self-citations that close a loop. Conjecture 1.3 is isolated as an external remaining ingredient. No quoted step in the provided text exhibits any of the six enumerated reduction patterns; the derivation chain remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; full manuscript required to audit.

pith-pipeline@v0.9.1-grok · 6435 in / 1089 out tokens · 36448 ms · 2026-06-27T23:24:19.767689+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 6 canonical work pages · 2 internal anchors

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[KMSW14] T. Kaletha, A. M´ ınguez, S. W. Shin and P.-J. White,Endoscopic classification of repre- sentations: inner forms of unitary groups. arXiv:1409.3731. ARITHMETIC WAVEFRONT SET AND HARISH-CHANDRA CHARACTER 37 [Kir04] A. A. Kirillov,Lectures on the orbit method. Graduate Studies in Mathematics, vol. 64, American Mathematical Society, Providence, RI,

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

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Lusztig and N

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[Mat87] H. Matumoto,Whittaker vectors and associated varieties, Invent. Math. 89 (1987), no. 1, 219–224. [Mat90] H. Matumoto,C −∞-Whittaker vectors for complex semisimple Lie groups, wave front sets, and Goldie rank polynomial representations. Ann. Sci. ´Ecole Norm. Sup. (4)23 (1990), no. 2, 311–367. [Mat92] H. Matumoto,C −∞-Whittaker vectors correspondin...

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[O15] Y. Oshima,On the restriction of Zuckerman’s derived functor modulesA q(λ)to reductive subgroups, Amer. J. Math. 137 (2015), no. 4, 1099–1138. [Prz90] T. Przebinda,The wave front set and the asymptotic support forp-adic groups. Pacific J. Math.141(1990), no. 2, 383–389. [Q14] Y. Qin,On the set of maximal nilpotent supports of supercuspidal representa...

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Math. Z. 277 (2014), no. 3-4, 1027–1048. [V78] D. Vogan,Gelfand-Kirillov dimension for Harish-Chandra modules. Invent. Math.48 (1978), no. 1, 75–98. [V79] D. Vogan,Irreducible characters of semisimple Lie groups. I,Duke Math. J. 46 (1979), no. 1, 61–108. [V87] D. Vogan,Unitary representations of reductive Lie groups. Ann. of Math. Stud., 118 Princeton Uni...

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Vogan,Associated varieties and unipotent representations

[V91] D. Vogan,Associated varieties and unipotent representations. Harmonic analysis on re- ductive groups, Proc. Conf., Brunswick/ME (USA) 1989, Progr. Math. 101 (1991), 315–

1989
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Academic Press, Inc., San Diego, CA, 1997, 243–339. [V00] D. Vogan,The method of coadjoint orbits for real reductive groups. IAS/Park City Math. Ser., 8 American Mathematical Society, Providence, RI, 2000, 179–238. [V17] D. Vogan,The size of infinite-dimensional representations. Jpn. J. Math. 12 (2017), no. 2, 175–210. [VZ84] D. Vogan and G. Zuckerman,Uni...

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[1] [1]

Adams and A

[AA24] J. Adams and A. Afgoustidis,Lowest K-types in the local Langlands correspondence, arXiv:2402.03552. [AA26] J. Adams and A. Afgoustidis,Nilpotent invariants for generic discrete series of real groups, Journal of the Institute of Mathematics of Jussieu. Published online 2026:1–23. doi:10.1017/S1474748026101650. [AC08] J. Adams and F. du Cloux, Algori...

work page doi:10.1017/s1474748026101650 2026

[2] [2]

Atobe and D

[AC26] H. Atobe and D. Ciubotaru,Endoscopic transfer and the wavefront upper bound conjec- ture. arXiv:2602.22504. [AGI+24] H. Atobe, W. T. Gan, A. Ichino, T. Kaletha, A. M´ ınguez and S. W. Shin,Local Inter- twining Relations and Co-temperedA-packets of Classical Groups, arXiv:2410.13504. [BB99] D. Barbasch and M. Boˇ ziˇ cevi´ c,The associated variety o...

work page doi:10.4171/jems/1609 1999

[3] [3]

[DMB25] D

xiv+186 pp. [DMB25] D. Davis and L. Mason-Brown,Hodge theory, intertwining functors, and the Orbit Method for real reductive groups. arXiv:2503.14794. [Duf77] M. Duflo,Sur la classification des id´ eaux primitifs dans l’alg` ebre enveloppante d’une alg` ebre de Lie semi-simple. Ann. of Math. (2) 105 (1977), no. 1, 107–120. [GGP12] W. T. Gan, B. Gross, and...

work page arXiv 1977

[4] [4]

Gourevitch and S

[GS15] D. Gourevitch and S. Sahi,Degenerate Whittaker functionals for real reductive groups. Amer. J. Math.137(2015), no. 2, 439–472. [GGS17] R. Gomez, D. Gourevitch, and S. Sahi,Generalized and degenerate Whittaker models. Compos. Math.153(2017), 223–256. [GGS21] R. Gomez, D. Gourevitch, and S. Sahi,Whittaker supports for representations of reductive gro...

2015

[5] [5]

Harris,Tempered representations and nilpotent orbits

[Hb12] B. Harris,Tempered representations and nilpotent orbits. Represent. Theory 16 (2012), 610–619. [Hb18] B. Harris,Wave front sets of reductive Lie group representations II. Trans. Amer. Math. Soc. 370 (2018), no. 8, 5931–5962. [HHO16] B. Harris, H. He, and G. ´Olafsson,Wave front sets of reductive Lie group representations. Duke Math. J. 165 (2016), ...

2012

[6] [6]

Losev and S

[LY23] I. Losev and S. Yu,On Harish-Chandra modules over quantizations of nilpotent orbits. arXiv:2309.11191. [J14] D. Jiang,Automorphic integral transforms for classical groups I: Endoscopy correspon- dences. Automorphic forms and related geometry: assessing the legacy of I. I. Piatetski- Shapiro, 179–242. Contemp. Math., 614 American Mathematical Societ...

work page arXiv 2014

[7] [7]

Endoscopic Classification of Representations: Inner Forms of Unitary Groups

[KMSW14] T. Kaletha, A. M´ ınguez, S. W. Shin and P.-J. White,Endoscopic classification of repre- sentations: inner forms of unitary groups. arXiv:1409.3731. ARITHMETIC WAVEFRONT SET AND HARISH-CHANDRA CHARACTER 37 [Kir04] A. A. Kirillov,Lectures on the orbit method. Graduate Studies in Mathematics, vol. 64, American Mathematical Society, Providence, RI,

work page internal anchor Pith review Pith/arXiv arXiv

[8] [8]

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

[Kos70] B. Kostant,Quantization and unitary representations,Lectures in Modern Analysis and Applications (C. Taam, eds.), Lecture Notes in Mathematics, vol. 170, Springer-Verlag, Berlin-Heidelberg-New York, 1970, pp. 87–208. [LLS24] B. Liu, C.-H. Lo and F. Shahidi,Jiang’s conjecture and fibers of the Barbasch-Vogan duality, Rad Hrvat. Akad. Znan. Umjet. M...

work page internal anchor Pith review Pith/arXiv arXiv 1970

[9] [9]

Lusztig and N

[LS79] G. Lusztig and N. Spaltenstein,Induced unipotent classes, J. Lond. Math. Soc. 19 (1979), 41–52. [MB25] L. Mason-Brown,Arthur’s Conjectures and the Orbit Method for Real Reductive Groups. The Langlands Program: I. Representations, Shimura Varieties, and Shtukas, 87–114. Proceedings of Symposia in Pure Mathematics Volume 112,

1979

[10] [10]

Matumoto,Whittaker vectors and associated varieties, Invent

[Mat87] H. Matumoto,Whittaker vectors and associated varieties, Invent. Math. 89 (1987), no. 1, 219–224. [Mat90] H. Matumoto,C −∞-Whittaker vectors for complex semisimple Lie groups, wave front sets, and Goldie rank polynomial representations. Ann. Sci. ´Ecole Norm. Sup. (4)23 (1990), no. 2, 311–367. [Mat92] H. Matumoto,C −∞-Whittaker vectors correspondin...

1987

[11] [11]

Oshima,On the restriction of Zuckerman’s derived functor modulesA q(λ)to reductive subgroups, Amer

[O15] Y. Oshima,On the restriction of Zuckerman’s derived functor modulesA q(λ)to reductive subgroups, Amer. J. Math. 137 (2015), no. 4, 1099–1138. [Prz90] T. Przebinda,The wave front set and the asymptotic support forp-adic groups. Pacific J. Math.141(1990), no. 2, 383–389. [Q14] Y. Qin,On the set of maximal nilpotent supports of supercuspidal representa...

2015

[12] [12]

Math. Z. 277 (2014), no. 3-4, 1027–1048. [V78] D. Vogan,Gelfand-Kirillov dimension for Harish-Chandra modules. Invent. Math.48 (1978), no. 1, 75–98. [V79] D. Vogan,Irreducible characters of semisimple Lie groups. I,Duke Math. J. 46 (1979), no. 1, 61–108. [V87] D. Vogan,Unitary representations of reductive Lie groups. Ann. of Math. Stud., 118 Princeton Uni...

2014

[13] [13]

Vogan,Associated varieties and unipotent representations

[V91] D. Vogan,Associated varieties and unipotent representations. Harmonic analysis on re- ductive groups, Proc. Conf., Brunswick/ME (USA) 1989, Progr. Math. 101 (1991), 315–

1989

[14] [14]

Academic Press, Inc., San Diego, CA, 1997, 243–339. [V00] D. Vogan,The method of coadjoint orbits for real reductive groups. IAS/Park City Math. Ser., 8 American Mathematical Society, Providence, RI, 2000, 179–238. [V17] D. Vogan,The size of infinite-dimensional representations. Jpn. J. Math. 12 (2017), no. 2, 175–210. [VZ84] D. Vogan and G. Zuckerman,Uni...

1997