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

Published 2 Apr 2026 in math.RT | (2604.01585v1)

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.

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Summary

The paper introduces a functorial theory of Bernstein–Zelevinsky derivatives to compute wavefront sets via iterated highest derivatives.
It extends Zelevinsky multisegment classification to KP and Savin covers with explicit formulas for Whittaker and semi-Whittaker model dimensions.
It establishes a precise link between wavefront sets, local Langlands correspondence, and Barbasch–Vogan duality in the setting of non-linear covering groups.

Wavefront Sets for Genuine Representations of $\mathrm{GL}$ -Covers of Kazhdan–Patterson or Savin Types

Introduction and Motivation

This work addresses the calculation and interpretation of wavefront sets for irreducible genuine representations of degree- $n$ Brylinski–Deligne (BD) central extensions of $p$ -adic general linear groups, focusing on the Kazhdan–Patterson (KP) and Savin covers. The critical innovations of the paper are:

A functorial theory of Bernstein–Zelevinsky (BZ) derivatives for such covering groups and an explicit computation of wavefront sets for irreducible genuine representations, expressed in terms of iterated highest derivatives.
A precise description of the relation between wavefront sets, Zelevinsky-type multisegments, and (in the KP case) the hypothetical local Langlands correspondence and covering Barbasch–Vogan duality.

The problem is highly nontrivial: genuine representations of covers can exhibit larger wavefront sets than their linear counterparts, leading to new phenomena requiring significant technical augmentation of the classical machinery.

Structure of BD-Covers and the Representation Theory Framework

Let $F$ be a $p$ -adic or large char- $p$ local field. The overarching structure starts with the $p$ -adic group $G_r = \mathrm{GL}_r(F)$ and its degree- $n$ BD central cover $\overline{G}_r$ :

$n$ 0

The cases of central interest are:

Kazhdan–Patterson covers: parameterized by a Weyl-invariant quadratic form with $n$ 1.
Savin covers: with $n$ 2 for simple coroots and $n$ 3 for $n$ 4.

In both cases, commutativity properties of block Levi subgroups simplify the representation theory sufficiently to allow an explicit BZ theory and Zelevinsky-type classification. The main technical challenge arises from possible non-commutativity for general BD covers, which the authors circumvent by restricting to the KP and Savin classes.

The representation category under consideration is that of genuine smooth complex representations for which the central $n$ 5 acts via a fixed faithful character.

Bernstein–Zelevinsky Derivatives and Zelevinsky Classification

The machinery of ordinary $n$ 6 representation theory, particularly the BZ derivatives and Zelevinsky multisegment parametrization, is extended to the covering group setting. There are two components:

Metaplectic/Extended Tensor Product: For blocks in Levi subgroups, the authors define and analyze both a (well-behaved) tensor product for Savin covers and a metaplectic tensor product (MTP), following [Mezo], [Takeda], and [Kaplan-Lapid-Zou], for KP covers. Associativity holds up to finite multiplicity, and central character compatibilities are managed by multifunctorial constructions.
Derivatives for Genuine Representations: Highest (iterated) BZ derivatives are defined for irreducible genuine representations. For segments and their Zelevinsky–type products, precise formulas for the dimension of Whittaker and semi-Whittaker models as well as for the structure of derivatives are proved. For instance, for a segment $n$ 7,

$n$ 8

where $n$ 9 and $p$ 0 are related combinatorially to $p$ 1 and the segment length. The formulas for Whittaker dimensions are explicit and depend (in the KP case) on the arithmetic of the covering degree and other data.

A main result is the exact calculation of the highest BZ derivative for any standard module attached to a multisegment, enabling the definition of a canonical partition $p$ 2 (the "wavefront partition") for each irreducible genuine representation.

Wavefront Set Computation and Main Theorem

The wavefront set $p$ 3 of an irreducible genuine representation is defined via the support of (degenerate) Whittaker functionals attached to nilpotent orbits. The Mœglin–Waldspurger theory is adapted to metaplectic covers using degenerate Whittaker models, following extensions by Gomez–Gourevitch–Sahi and Patel.

The main theorem is:

If $p$ 4 is the standard module attached to the multisegment $p$ 5 for a KP or Savin cover, then the wavefront set $p$ 6 is a singleton containing the partition $p$ 7 determined explicitly by iterated highest BZ derivatives.

In formulas, $p$ 8

This generalizes the classical Zelevinsky/Mœglin–Waldspurger results to the non-linear setting and is essentially optimal under current knowledge. The authors also give explicit combinatorial criteria for the nonvanishing of (semi-)Whittaker models—the dimensions of which are significantly larger than in the linear case.

A corollary is that in the KP and Savin cases, representations with “small enough” segment data are always generic (i.e., admit Whittaker functionals). Strong multiplicity one fails: the dimensions of generic models can be arbitrarily large.

Compatibility with Local Langlands Correspondence and Barbasch–Vogan Duality

For KP covers, the authors recover and refine expected connections with the hypothetical covering group local Langlands correspondence and the Barbasch–Vogan duality for orbits:

By using the metaplectic correspondence (via Flicker–Kazhdan–Patterson) between KP-cover discrete series and linear $p$ 9 discrete series, explicit matching of L-parameters and central characters is established.
The explicit wavefront set formula is shown to be equivalent to the partition induced from the image of the Barbasch–Vogan duality applied to the nilpotent orbit attached (via the dual group and LLC) to the Aubert–Zelevinsky dual of the original standard module.

Concretely,

$F$ 0

where the Barbasch–Vogan map $F$ 1 is made explicit and the nilpotent orbit is read off from the L-parameter constructed from the metaplectic lifting.

This gives a precise and new form of the expected LLC–wavefront set relation for covers.

Implications, Technical Innovations, and Outlook

The technical achievements of the paper are manifold:

Extension of BZ and Zelevinsky multisegment theory to multiblock situation in metaplectic and Savin covers, including nonblock-compatibility in the KP case.
Multiplicity formulae for generic and degenerate models, showing that dimensions for metaplectic covers can far exceed the corresponding linear case.
Complete wavefront set calculation for irreducible standard modules in KP and Savin covers, together with compatibility with the (covering) Barbasch–Vogan duality and conjectural LLC.

Practically, these results provide a firm theoretical underpinning for harmonic analysis on metaplectic covers of $F$ 2 and thus for related automorphic representation theory and number theory (theta correspondences, period integrals, and beyond).

Theoretically, the methods illuminate the extent to which classical ideas from linear $F$ 3-adic groups hold in the non-linear context, and highlight precisely where new phenomena and obstructions arise. The explicit connection to LLC and orbit duality is especially germane for further advances in the theory of covering groups.

Future Developments: The results and techniques offer a prototype for the investigation and calculation of wavefront sets, Whittaker models, and duality phenomena for more general BD covers and other reductive groups, and strongly suggest precise forms for the LLC and orbit duality conjectures in the nonlinear (covering) setting. The extension to wild ramification and nonadmissible coverings, as well as to higher rank covering groups, remains a substantial open line of inquiry.

Conclusion

The paper provides an explicit, detailed theory of wavefront sets for irreducible genuine representations of degree- $F$ 4 metaplectic and Savin type covers of $F$ 5, establishing their calculation in terms of highest BZ derivatives and Zelevinsky multisegments. It demonstrates compatibility with conjectural LLC and Barbasch–Vogan duality and exhibits the intricate yet highly structured nature of both the generic and degenerate model theory for such nonlinear representations. The methods and results significantly advance the explicit harmonic analysis of covering groups and clarify the structure of their genuine representation categories (2604.01585).