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Covering Barbasch-Vogan duality and wavefront sets of genuine representations

Fan Gao, Baiying Liu, Chi-Heng Lo, Freydoon Shahidi

TL;DR

This work introduces a covering Barbasch–Vogan duality $d_{BV,G}^{(n)}$ for Brylinski–Deligne covers and proves its key structural properties, including order-reversing behavior and Levi-induction compatibility. Using this duality, the authors formulate a generalized upper bound for the geometric wavefront sets ${ m WF}^{ m geo}({ m AZ}( au))$ of genuine representations in terms of $d_{BV,G}^{(n)}( ext{O}( ext{phi}_ au))$, with equality in tempered contexts. They establish the bound for Kazhdan–Patterson covers of ${ m GL}_r$ and provide explicit combinatorial realizations for classical groups, as well as evidence from theta representations and unramified cases to support the conjecture. The paper also outlines desiderata for a local Langlands correspondence for covers and shows how the general conjecture reduces to discrete/anti-tempered cases via a principled reduction framework, thereby charting a path toward a complete understanding of wavefront sets in genuine p-adic representation theory. Altogether, these results deepen the interplay between nilpotent orbits, L-parameters, and wavefront phenomena in the setting of covering groups, with concrete progress in the GL_s(K) regime and broader implications for automorphic descent and related conjectures.

Abstract

In this paper, we start by defining a covering Barbasch-Vogan duality and prove some of its properties. Then, for genuine representations of $p$-adic covering groups we formulate an upper bound conjecture for their wavefront sets using this covering Barbasch-Vogan duality and reduce it to anti-discrete representations. The formulation generalizes that of Ciubotaru-Kim and Hazeltine-Liu-Lo-Shahidi for linear algebraic groups. We prove this upper bound conjecture for Kazhdan-Patterson coverings of general linear groups.

Covering Barbasch-Vogan duality and wavefront sets of genuine representations

TL;DR

This work introduces a covering Barbasch–Vogan duality for Brylinski–Deligne covers and proves its key structural properties, including order-reversing behavior and Levi-induction compatibility. Using this duality, the authors formulate a generalized upper bound for the geometric wavefront sets of genuine representations in terms of , with equality in tempered contexts. They establish the bound for Kazhdan–Patterson covers of and provide explicit combinatorial realizations for classical groups, as well as evidence from theta representations and unramified cases to support the conjecture. The paper also outlines desiderata for a local Langlands correspondence for covers and shows how the general conjecture reduces to discrete/anti-tempered cases via a principled reduction framework, thereby charting a path toward a complete understanding of wavefront sets in genuine p-adic representation theory. Altogether, these results deepen the interplay between nilpotent orbits, L-parameters, and wavefront phenomena in the setting of covering groups, with concrete progress in the GL_s(K) regime and broader implications for automorphic descent and related conjectures.

Abstract

In this paper, we start by defining a covering Barbasch-Vogan duality and prove some of its properties. Then, for genuine representations of -adic covering groups we formulate an upper bound conjecture for their wavefront sets using this covering Barbasch-Vogan duality and reduce it to anti-discrete representations. The formulation generalizes that of Ciubotaru-Kim and Hazeltine-Liu-Lo-Shahidi for linear algebraic groups. We prove this upper bound conjecture for Kazhdan-Patterson coverings of general linear groups.

Paper Structure

This paper contains 10 sections, 29 theorems, 319 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Acknowledgement
  4. Covering Barbasch--Vogan duality
  5. Conjecture on the wavefront sets for covers
  6. Order reversing
  7. Some evidence for the conjecture
  8. Reduction of the conjecture for general types
  9. Desiderata for local Langlands correspondence
  10. Reduction of the conjecture

Key Result

Theorem 1.1

The covering Barbasch--Vogan duality $d_{BV,G}^{(n)}:\mathcal{N}(\overline{G}^\vee) \to \mathcal{N}({\bf G})$ is explicitly given in § SS:fix-cov for covers of $G$ of classical type and in Appendix A:dBVexc for exceptional type. Moreover, it satisfies the following properties:

Theorems & Definitions (63)

  • Theorem 1.1
  • Conjecture 1.2
  • Theorem 1.3
  • Definition 2.1
  • Conjecture 2.2
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Theorem 3.3
  • Lemma 3.4
  • ...and 53 more