A degenerate Whittaker criterion for $\mathrm GL_{2n}$

Taiwang Deng

arxiv: 2607.01598 · v1 · pith:VBTWIIVUnew · submitted 2026-07-02 · 🧮 math.RT

A degenerate Whittaker criterion for mathrm GL_(2n)

Taiwang Deng This is my paper

Pith reviewed 2026-07-03 03:40 UTC · model grok-4.3

classification 🧮 math.RT MSC 22E50

keywords degenerate Whittaker modelsZelevinsky classificationtwisted Jacquet modulesGL(2n)Langlands classificationadjoint L-functionswave-front sets

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The pith

For representations of GL_{2n}, the twisted Jacquet module vanishes if and only if the Zelevinsky dual contains a segment of length at least n+1.

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

The paper translates an orbit-theoretic criterion for the vanishing of twisted Jacquet modules into the Langlands-Zelevinsky classification for irreducible representations of GL_{2n} over non-Archimedean local fields. It shows that for π given by multisegment m, the module π_{N,ψ} is zero exactly when the dual m^t has a segment longer than n. This gives a concrete combinatorial test and partially resolves a conjecture of Prasad relating the vanishing to poles of the adjoint L-function, while providing counterexamples to the full conjecture.

Core claim

The central claim is that if π = L(m) is an irreducible admissible representation of GL_{2n}(F), then the twisted Jacquet module π_{N,ψ} vanishes if and only if the Zelevinsky dual m^t contains a segment of length at least n+1. This equivalence follows from translating the wave-front set condition of Gomez-Gourevitch-Sahi into the language of multisegments.

What carries the argument

The Zelevinsky dual m^t of the multisegment m parametrizing the representation, whose maximal segment length controls the vanishing of the twisted Jacquet module.

If this is right

Vanishing implies the pole conditions on the adjoint L-function L(s, π × π^∨) predicted by Prasad.
For principal series containing the trivial representation, the vanishing can be determined explicitly for each generalized Steinberg constituent.
The criterion holds for all irreducible admissible representations of GL_{2n}(F).

Where Pith is reading between the lines

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

The combinatorial condition may allow direct computation of vanishing without computing wave-front sets for other groups.
Counterexamples to Prasad's conjecture suggest that L-function poles are necessary but not sufficient for non-vanishing in some cases.

Load-bearing premise

The orbit-theoretic vanishing criterion can be translated directly into the Langlands-Zelevinsky classification without additional restrictions on the representation or the field.

What would settle it

Finding an irreducible representation of GL_{2n} where the Zelevinsky dual has no segment of length n+1 or more but the twisted Jacquet module still vanishes, or the opposite case.

read the original abstract

Let $F$ be a non-Archimedean local field. Let $N$ be the unipotent radical of the standard parabolic subgroup of $\mathrm GL_{2n}(F)$ of type $(n,n)$ with fixed nondegenerate additive character $\psi$. For an irreducible admissible representation $\pi$ of $\mathrm GL_{2n}(F)$, a theorem due to Gomez--Gourevitch--Sahi on generalized Whittaker models gives a criterion for the vanishing of the twisted Jacquet module $\pi_{N,\psi}$ in terms of the wave-front set. We translate this orbit-theoretic answer into Langlands--Zelevinsky data: if $\pi=L(\mathfrak m)$, then $\pi_{N,\psi}=0$ if and only if the Zelevinsky dual $\mathfrak m^{\mathrm t}$ contains a segment of length at least $n+1$. We do this in response to a conjecture proposed by D.Prasad about the vanishing of $\pi_{N,\psi}$ in terms of the adjoint $L$-function $L(s,\pi\times\pi^\vee)$. We prove that, for every irreducible representation $\pi$, vanishing of $\pi_{N,\psi}$ implies the pole inequalities predicted by D.Prasad. However, we show that the converse implication is false by an explicit counterexample for $\mathrm GL_4(F)$. For the generalized Steinberg constituents $v_{P_\beta}^G$ of the principal series containing the trivial representation, we make an explicit calculation of when $\pi_{N,\psi}$ is zero. In particular, for $\mathrm GL_6(F)$, exactly three of the $32$ constituents of such a principal series violate the converse direction of the conjecture proposed by D.Prasad.

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

The paper translates the GGS orbit criterion into a clean segment-length condition on Zelevinsky data and supplies explicit counterexamples showing the converse of Prasad's L-function conjecture fails on GL4 and GL6.

read the letter

The main things to know are the explicit translation of the Gomez-Gourevitch-Sahi vanishing criterion into Zelevinsky language and the concrete counterexamples to the converse direction of Prasad's conjecture.

The paper states that for π = L(m) on GL_{2n}(F), the twisted Jacquet module vanishes exactly when the dual multisegment m^t contains a segment of length at least n+1. It proves the vanishing direction implies the expected adjoint L-function pole conditions, then gives counterexamples on GL4 where the L-function condition holds but the module does not vanish. For the 32 generalized Steinberg constituents of a certain principal series on GL6, exactly three violate the converse.

The translation step and the explicit checks are the useful parts. They turn an orbit-theoretic statement into something directly usable with Langlands-Zelevinsky parameters and show the directionality of the implication with small-rank calculations that can be inspected.

The work rests on the cited GGS theorem, which is standard, and the translation appears to go through without extra restrictions on the field or the representation. The counterexamples are limited to n=2 and n=3, which is enough to disprove a general converse but leaves the general picture open. No internal inconsistencies show up in the stated claims.

This is for people working on Whittaker models and L-function conjectures for classical groups. A reader who needs a usable criterion in Zelevinsky data or wants to see where Prasad-type predictions break will find it directly applicable.

It deserves a serious referee. The statements are precise and the counterexamples are concrete enough to check.

Referee Report

2 major / 2 minor

Summary. The paper translates the Gomez--Gourevitch--Sahi orbit-theoretic criterion for vanishing of the twisted Jacquet module π_{N,ψ} on GL_{2n}(F) into Langlands--Zelevinsky data: for an irreducible admissible π = L(m), π_{N,ψ} vanishes if and only if the dual multisegment m^t contains a segment of length at least n+1. It proves that this vanishing implies the adjoint L-function pole inequalities conjectured by Prasad, but constructs explicit counterexamples showing the converse fails, including one on GL_4(F) and three of the 32 generalized Steinberg constituents on GL_6(F).

Significance. The combinatorial criterion in terms of Zelevinsky segments supplies an explicit, computable test for degenerate Whittaker models that builds directly on the cited GGS theorem and standard facts about L-functions. The counterexamples on low-rank groups clarify the precise relationship to Prasad's conjecture and include concrete calculations that can be independently verified. The work avoids circularity and provides falsifiable predictions via the explicit multisegment data.

major comments (2)

[Main translation theorem] The translation of the GGS wavefront-set condition into the segment-length criterion (stated in the abstract and proved in the main theorem) assumes the correspondence holds without additional restrictions for all irreducible admissible representations; the manuscript should explicitly verify this for non-tempered cases or cite the precise step where the orbit dimension maps to segment length ≥ n+1.
[Counterexample section (GL_4)] For the GL_4(F) counterexample, the explicit multisegment m and the computation confirming that L(s, π × π^∨) has the predicted pole while π_{N,ψ} does not vanish should be expanded to include the precise Zelevinsky data and the L-function order calculation.

minor comments (2)

[Introduction] Notation for the Zelevinsky dual m^t and the segment length should be introduced with a brief reminder in the introduction for readers unfamiliar with the classification.
[GL_6 calculation] The count of 32 constituents on GL_6(F) and the identification of the three violating ones would benefit from a short table or explicit list of the multisegments.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the recommendation of minor revision. We address each major comment below.

read point-by-point responses

Referee: The translation of the GGS wavefront-set condition into the segment-length criterion (stated in the abstract and proved in the main theorem) assumes the correspondence holds without additional restrictions for all irreducible admissible representations; the manuscript should explicitly verify this for non-tempered cases or cite the precise step where the orbit dimension maps to segment length ≥ n+1.

Authors: The GGS theorem is stated for all irreducible admissible representations, and the Zelevinsky classification provides a bijection between multisegments and irreducible representations that holds uniformly, including for non-tempered representations. The mapping from wavefront set (via orbit dimension) to the existence of a segment of length at least n+1 in m^t follows directly from the standard combinatorial description of the wavefront set in terms of Zelevinsky data. We will add a clarifying remark after the statement of the main theorem, citing the relevant properties of the Zelevinsky classification that ensure the correspondence applies without further restrictions. revision: yes
Referee: For the GL_4(F) counterexample, the explicit multisegment m and the computation confirming that L(s, π × π^∨) has the predicted pole while π_{N,ψ} does not vanish should be expanded to include the precise Zelevinsky data and the L-function order calculation.

Authors: We agree that providing the explicit multisegment data and the detailed order computation for the adjoint L-function will strengthen the presentation of the counterexample. We will expand the relevant section (currently containing the GL_4 example) to include the precise Zelevinsky multisegment m together with the explicit calculation of the pole order of L(s, π × π^∨). revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central result translates an external orbit-theoretic criterion from Gomez--Gourevitch--Sahi into Zelevinsky multisegment language for GL_{2n}. This relies on standard representation-theoretic facts and explicit counterexamples to Prasad's conjecture, none of which reduce by definition or self-citation to the paper's own fitted quantities or inputs. No self-definitional steps, fitted predictions, or load-bearing self-citations appear in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claims rest on the cited Gomez--Gourevitch--Sahi theorem, the Langlands-Zelevinsky classification, and standard properties of adjoint L-functions; no new free parameters or invented entities are introduced.

axioms (3)

standard math Gomez--Gourevitch--Sahi theorem on generalized Whittaker models
The paper translates this theorem into Zelevinsky data.
standard math Langlands-Zelevinsky classification of irreducible admissible representations of GL_{2n}(F)
Used to express the vanishing criterion in terms of multisegments.
standard math Existence and basic properties of adjoint L-functions and their poles
Used to formulate and partially resolve Prasad's conjecture.

pith-pipeline@v0.9.1-grok · 5851 in / 1493 out tokens · 50590 ms · 2026-07-03T03:40:38.397294+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 1 canonical work pages

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Taiwang Deng. Study of multiplicities in induced representations of $GL_n$ through a symmetric reduction. Manuscr. Math. , 171(1-2):23--72, 2023

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Generalized and degenerate Whittaker models

Raul Gomez, Dmitry Gourevitch, and Siddhartha Sahi. Generalized and degenerate Whittaker models. Compos. Math. , 153(2):223--256, 2017

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Harshitha and C

C. Harshitha and C. G. Venketasubramanian. Structure of twisted Jacquet modules of principal series representations of GL_ 2n (F) . Preprint, arXiv :2512.24737 [math. RT ] (2026), 2026

work page arXiv 2026
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On the local and global exterior square $L$ -functions of $GL_n$

Pramod Kumar Kewat and Ravi Raghunathan. On the local and global exterior square $L$ -functions of $GL_n$ . Math. Res. Lett. , 19(4):785--804, 2012

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Linear and Shalika local periods for the mirabolic group, and some consequences

Nadir Matringe. Linear and Shalika local periods for the mirabolic group, and some consequences. J. Number Theory , 138:1--19, 2014

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Moeglin and J

C. Moeglin and J. L. Waldspurger. Mod \`e les de Whittaker d \'e g \'e n \'e res pour des groupes p-adiques. ( Degenerate Whittaker models of p-adic groups). Math. Z. , 196:427--452, 1987

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A. V. Zelevinsky. Induced representations of reductive $p$ -adic groups. II : On irreducible representations of $GL(n)$ . Ann. Sci. \'E c. Norm. Sup \'e r. (4) , 13:165--210, 1980

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[1] [1]

The endoscopic classification of representations

James Arthur. The endoscopic classification of representations. Orthogonal and symplectic groups , volume 61 of Colloq. Publ., Am. Math. Soc. Providence, RI: American Mathematical Society (AMS), 2013

2013

[2] [2]

Automorphic L -functions

Armand Borel. Automorphic L -functions. Automorphic forms, representations and L -functions, Proc . Symp . Pure Math . Am . Math . Soc ., Corvallis / Oregon 1977, Proc . Symp . Pure Math . 33, 2, 27-61 (1979)., 1979

1977

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I. N. Bernstein and A. V. Zelevinsky. Induced representations of reductive \(p\) -adic groups. I . Ann. Sci. \'E c. Norm. Sup \'e r. (4) , 10:441--472, 1977

1977

[4] [4]

Study of multiplicities in induced representations of \(GL_n\) through a symmetric reduction

Taiwang Deng. Study of multiplicities in induced representations of \(GL_n\) through a symmetric reduction. Manuscr. Math. , 171(1-2):23--72, 2023

2023

[5] [5]

Generalized and degenerate Whittaker models

Raul Gomez, Dmitry Gourevitch, and Siddhartha Sahi. Generalized and degenerate Whittaker models. Compos. Math. , 153(2):223--256, 2017

2017

[6] [6]

Harshitha and C

C. Harshitha and C. G. Venketasubramanian. Structure of twisted Jacquet modules of principal series representations of GL_ 2n (F) . Preprint, arXiv :2512.24737 [math. RT ] (2026), 2026

work page arXiv 2026

[7] [7]

On the local and global exterior square \(L\) -functions of \(GL_n\)

Pramod Kumar Kewat and Ravi Raghunathan. On the local and global exterior square \(L\) -functions of \(GL_n\) . Math. Res. Lett. , 19(4):785--804, 2012

2012

[8] [8]

Linear and Shalika local periods for the mirabolic group, and some consequences

Nadir Matringe. Linear and Shalika local periods for the mirabolic group, and some consequences. J. Number Theory , 138:1--19, 2014

2014

[9] [9]

Moeglin and J

C. Moeglin and J. L. Waldspurger. Mod \`e les de Whittaker d \'e g \'e n \'e res pour des groupes p-adiques. ( Degenerate Whittaker models of p-adic groups). Math. Z. , 196:427--452, 1987

1987

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Classes unipotentes et sous-groupes de Borel , volume 946 of Lect

Nicolas Spaltenstein. Classes unipotentes et sous-groupes de Borel , volume 946 of Lect. Notes Math. Springer, Cham, 1982

1982

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A. V. Zelevinsky. Induced representations of reductive \(p\) -adic groups. II : On irreducible representations of \(GL(n)\) . Ann. Sci. \'E c. Norm. Sup \'e r. (4) , 13:165--210, 1980

1980