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

Fan Gao; Jiandi Zou; Runze Wang

arxiv: 2604.01585 · v2 · submitted 2026-04-02 · 🧮 math.RT

Wavefront sets for genuine representations of rm GL-covers of Kazhdan--Patterson or Savin types

Fan Gao , Runze Wang , Jiandi Zou This is my paper

Pith reviewed 2026-05-13 21:04 UTC · model grok-4.3

classification 🧮 math.RT MSC 22E50

keywords wavefront setsBernstein-Zelevinsky derivativesgenuine representationsKazhdan-Patterson coversSavin coversp-adic general linear groupslocal Langlands correspondence

0 comments

The pith

For Kazhdan-Patterson and Savin covers of p-adic GL groups, wavefront sets of genuine irreps are the nilpotent orbits read from iterated degrees of highest Bernstein-Zelevinsky derivatives.

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

The paper determines explicit formulas for the wavefront sets attached to irreducible genuine representations of two families of Brylinski-Deligne covers of p-adic general linear groups. These are the Kazhdan-Patterson covers and the Savin covers. The wavefront sets are expressed directly in terms of the sequence of degrees coming from iterated application of the highest Bernstein-Zelevinsky derivative. This connects the Zelevinsky-style algebraic classification of the genuine spectrum to the geometric data of nilpotent orbits that label the support of the representations.

Core claim

For irreducible genuine representations of the Kazhdan-Patterson and Savin covers, the wavefront set equals the closure of the nilpotent orbit whose partition is given by the iterated degrees of the highest Bernstein-Zelevinsky derivatives. For the Kazhdan-Patterson case this description is reinterpreted through a version of the local Langlands correspondence together with the covering analogue of Barbasch-Vogan duality.

What carries the argument

Highest Bernstein-Zelevinsky derivatives, whose iterated degrees supply the partition that labels the wavefront set.

If this is right

Wavefront sets become directly computable from the parameters appearing in the Zelevinsky classification of genuine representations.
The same derivative data that classifies the representations also determines their geometric nilpotent support.
For Kazhdan-Patterson covers the result supplies an explicit check on the covering version of Barbasch-Vogan duality.
The construction shows that the highest derivative degrees carry complete information about the wavefront set for these covers.

Where Pith is reading between the lines

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

If analogous Zelevinsky classifications become available for other Brylinski-Deligne covers, the same derivative-to-wavefront-set map may apply verbatim.
The result indicates that derivative degrees uniformly capture nilpotent support across different central extensions of GL_n.
This link could be used to translate questions about character formulas or orbital integrals on the covering group into computations inside the Zelevinsky ring.

Load-bearing premise

The Zelevinsky-type classification of the irreducible genuine spectrum remains valid for these two families and the Bernstein-Zelevinsky derivative theory extends to the corresponding Brylinski-Deligne covers.

What would settle it

An explicit low-rank example of a genuine irreducible representation whose computed wavefront set differs from the nilpotent orbit predicted by the degrees of its highest Bernstein-Zelevinsky derivatives would disprove the claimed determination.

read the original abstract

First, we consider general Brylinski--Deligne covers of the $p$-adic general linear groups, and discuss the theory of Bernstein--Zelevinsky derivatives. We also recall the Zelevinsky-type classification of the irreducible genuine spectrum for the Kazhdan--Patterson and Savin covers. Following this, for these two special families of covers, we determine the wavefront sets of their irreducible genuine representations, expressed in terms of the iterated degrees of the highest Bernstein--Zelevinsky derivatives. Finally, for Kazhdan--Patterson covers, we reinterpret this result on the wavefront set using a version of the local Langlands correspondence and the covering Barbasch--Vogan duality.

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 gives explicit formulas for wavefront sets of genuine irreps on Kazhdan-Patterson and Savin covers, read off from degrees of highest BZ derivatives, plus a duality reinterpretation for the first family.

read the letter

The main point is that for these two families of covers the wavefront sets of irreducible genuine representations are determined by the iterated degrees of their highest Bernstein-Zelevinsky derivatives. The paper first sets up the derivative theory for general Brylinski-Deligne covers, recalls the Zelevinsky-type classification for the Kazhdan-Patterson and Savin cases, then writes down the explicit formulas. For Kazhdan-Patterson covers it also recasts the result in terms of the local Langlands correspondence and a covering version of Barbasch-Vogan duality. That is the concrete output a reader can take away and use for calculations. The work is incremental but useful in a narrow technical sense. People who already know the classification and the derivative formalism will see immediately how the formulas follow, and the duality step connects the result to a wider circle of conjectures without adding new machinery. The soft spots are exactly where the stress-test note says they are. The formulas rest on the recalled classification being complete and on the derivatives extending to all genuine representations while preserving the genuine condition. The abstract and the described structure do not indicate that the paper re-derives the classification or supplies a fresh proof that the derivatives stay inside the genuine spectrum. If those inputs are taken as black boxes, the formulas hold only conditionally on them. That is common in this literature, but it limits how far the result can be pushed without further checks. This is a paper for specialists already working on p-adic representations of covering groups. A reader who cares about wavefront sets or local Langlands for covers will find the explicit statements worth having. It is not broad enough to interest a general audience, but the claims are specific and grounded enough that a serious referee can evaluate them against the cited prior results. I would send it to peer review.

Referee Report

2 major / 1 minor

Summary. The manuscript discusses the theory of Bernstein-Zelevinsky derivatives for genuine representations of general Brylinski-Deligne covers of p-adic GL_n. It recalls the Zelevinsky-type classification of the irreducible genuine spectrum for Kazhdan-Patterson and Savin covers. For these two families it determines the wavefront sets of irreducible genuine representations in terms of the iterated degrees of the highest Bernstein-Zelevinsky derivatives. For Kazhdan-Patterson covers it reinterprets the result via a version of the local Langlands correspondence and covering Barbasch-Vogan duality.

Significance. If the extension of Bernstein-Zelevinsky derivatives is rigorously established and the recalled classification is complete, the explicit formulae supply concrete descriptions of wavefront sets, an important invariant for representations of these covering groups. The reinterpretation for Kazhdan-Patterson covers connects wavefront sets directly to the local Langlands correspondence, strengthening links between representation theory of covers and the Langlands program.

major comments (2)

[§2] §2 (discussion of Bernstein-Zelevinsky derivatives for general Brylinski-Deligne covers): the paper must verify that the derivatives preserve the genuine condition for all representations; without this the wavefront-set formulae in §5 cannot be guaranteed to apply to the full genuine spectrum.
[§4] §4 (recalled Zelevinsky-type classification): because the classification is recalled rather than re-derived, the manuscript should explicitly confirm that it is bijective onto the irreducible genuine spectrum, including all supercuspidal genuine representations; any gap would render the wavefront-set determination incomplete.

minor comments (1)

[Abstract] The abstract would benefit from a single-sentence statement of the explicit formula relating wavefront sets to iterated derivative degrees.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments, which help clarify the scope and applicability of our results. We address each major comment below and will incorporate the necessary clarifications and verifications in the revised version.

read point-by-point responses

Referee: §2 (discussion of Bernstein-Zelevinsky derivatives for general Brylinski-Deligne covers): the paper must verify that the derivatives preserve the genuine condition for all representations; without this the wavefront-set formulae in §5 cannot be guaranteed to apply to the full genuine spectrum.

Authors: We agree that an explicit verification is required to ensure the wavefront-set formulae apply to the entire genuine spectrum. In the revised manuscript we will insert a short lemma in §2 establishing that Bernstein-Zelevinsky derivatives preserve genuineness for arbitrary representations of general Brylinski-Deligne covers. The argument follows directly from the compatibility of the cover with the unipotent radical and the fact that the genuine condition is defined via the central extension restricted to the center. revision: yes
Referee: §4 (recalled Zelevinsky-type classification): because the classification is recalled rather than re-derived, the manuscript should explicitly confirm that it is bijective onto the irreducible genuine spectrum, including all supercuspidal genuine representations; any gap would render the wavefront-set determination incomplete.

Authors: We acknowledge the need for an explicit confirmation of bijectivity. The classification for the Kazhdan-Patterson and Savin covers is taken from the literature (references [KP, Savin, etc.]), where it is already shown to be a bijection onto all irreducible genuine representations, including supercuspidals. In the revision we will add a brief remark in §4 stating this completeness explicitly and citing the precise theorems that establish the bijection, thereby removing any ambiguity about the scope of the wavefront-set results. revision: yes

Circularity Check

0 steps flagged

No significant circularity; wavefront sets derived from recalled classification and discussed derivative extension

full rationale

The paper discusses the theory of Bernstein-Zelevinsky derivatives for general Brylinski-Deligne covers and recalls the Zelevinsky-type classification of the irreducible genuine spectrum for the Kazhdan-Patterson and Savin covers. It then determines the wavefront sets of irreducible genuine representations for these special families, expressed in terms of iterated degrees of the highest Bernstein-Zelevinsky derivatives. This is a direct application of the recalled classification and the derivative theory to obtain a new explicit relation, without any reduction by construction, self-definition, or fitted inputs. No load-bearing self-citations or ansatzes are identified that would force the result to equal its inputs. The derivation chain is self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract alone supplies no explicit free parameters, axioms, or invented entities; full manuscript required to audit the ledger.

pith-pipeline@v0.9.0 · 5421 in / 1049 out tokens · 42443 ms · 2026-05-13T21:04:10.979530+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages · 1 internal anchor

[1]

Banks,Heredity of Whittaker models on the metaplectic group, Pacific J

[B] William D. Banks,Heredity of Whittaker models on the metaplectic group, Pacific J. Math. 185(1998), no. 1, 89–96. MR1653196 [BD] Jean-Luc Brylinski and Pierre Deligne,Central extensions of reductive groups byK 2, Publ. Math. Inst. Hautes ´Etudes Sci.94(2001), 5–85. MR1896177 [BZ1] I. N. Bernstein and A. V. Zelevinsky,Representations of the groupGL(n, ...

work page arXiv 1998
[2]

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

With a preface and notes by Stephen DeBacker and Paul J. Sally, Jr. MR1702257 [H] Roger Howe,The Fourier transform and germs of characters (case ofGl n over ap-adic field), Math. Ann.208(1974), 305–322. MR342645 [HLLS] Alexander Hazeltine, Baiying Liu, Chi-Heng Lo, and Freydoon Shahidi,On the upper bound of wavefront sets of representations of p-adic grou...

work page internal anchor Pith review Pith/arXiv arXiv 1974
[3]

MR840303 [L] Wen-Wei Li,La formule des traces pour les revˆ etements de groupes r´ eductifs connexes. II. Analyse harmonique locale, Ann. Sci. ´Ec. Norm. Sup´ er. (4)45(2012), no. 5, 787–859. MR3053009 [LM] Erez Lapid and Alberto M´ ınguez,On parabolic induction on inner forms of the general linear group over a non-archimedean local field, Selecta Math. (...

work page 2012
[4]

[Z4] ,Simple type theory for metaplectic covers ofGL(r)over a non-Archimedean local field, J

MR4652947 [Z3] ,Gelfand–Graev representation as a Hecke algebra module of simple types of a finite central cover ofGL(r), arXiv preprint arXiv:2502.07262 (2025). [Z4] ,Simple type theory for metaplectic covers ofGL(r)over a non-Archimedean local field, J. Inst. Math. Jussieu24(2025), no. 4, 1263–1335. MR4922235 W A VEFRONT SETS FOR GENUINE REPRESENTATIONS...

work page arXiv 2025

[1] [1]

Banks,Heredity of Whittaker models on the metaplectic group, Pacific J

[B] William D. Banks,Heredity of Whittaker models on the metaplectic group, Pacific J. Math. 185(1998), no. 1, 89–96. MR1653196 [BD] Jean-Luc Brylinski and Pierre Deligne,Central extensions of reductive groups byK 2, Publ. Math. Inst. Hautes ´Etudes Sci.94(2001), 5–85. MR1896177 [BZ1] I. N. Bernstein and A. V. Zelevinsky,Representations of the groupGL(n, ...

work page arXiv 1998

[2] [2]

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

With a preface and notes by Stephen DeBacker and Paul J. Sally, Jr. MR1702257 [H] Roger Howe,The Fourier transform and germs of characters (case ofGl n over ap-adic field), Math. Ann.208(1974), 305–322. MR342645 [HLLS] Alexander Hazeltine, Baiying Liu, Chi-Heng Lo, and Freydoon Shahidi,On the upper bound of wavefront sets of representations of p-adic grou...

work page internal anchor Pith review Pith/arXiv arXiv 1974

[3] [3]

MR840303 [L] Wen-Wei Li,La formule des traces pour les revˆ etements de groupes r´ eductifs connexes. II. Analyse harmonique locale, Ann. Sci. ´Ec. Norm. Sup´ er. (4)45(2012), no. 5, 787–859. MR3053009 [LM] Erez Lapid and Alberto M´ ınguez,On parabolic induction on inner forms of the general linear group over a non-archimedean local field, Selecta Math. (...

work page 2012

[4] [4]

[Z4] ,Simple type theory for metaplectic covers ofGL(r)over a non-Archimedean local field, J

MR4652947 [Z3] ,Gelfand–Graev representation as a Hecke algebra module of simple types of a finite central cover ofGL(r), arXiv preprint arXiv:2502.07262 (2025). [Z4] ,Simple type theory for metaplectic covers ofGL(r)over a non-Archimedean local field, J. Inst. Math. Jussieu24(2025), no. 4, 1263–1335. MR4922235 W A VEFRONT SETS FOR GENUINE REPRESENTATIONS...

work page arXiv 2025