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

Fan Gao, Runze Wang, Jiandi Zou

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.

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

Abstract

First, we consider general Brylinski--Deligne covers of the -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.

Paper Structure

This paper contains 32 sections, 33 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main result
  4. Acknowledgement
  5. Extended metaplectic tensor product and Whittaker models
  6. Covers of $G_r$
  7. Finite central covers
  8. Finite central covers of $G_r$
  9. Metaplectic tensor product
  10. Tensor product for S-covers
  11. Metaplectic tensor product for KP-covers
  12. Extended metaplectic tensor product
  13. Whittaker and semi-Whittaker models
  14. Bernstein--Zelevinsky theory
  15. Extended Bernstein-Zelevinsky product
  16. ...and 17 more sections

Key Result

Theorem 1.1

Let $\overline{G_r}$ be an $n$-fold cover of $G_r$ of Savin type or Kazhdan--Patterson type. Assume $p\nmid n$. Then for every $Z(\mathfrak{m}) \in {\rm Irr}_{\epsilon} (\overline{G_r})$, one has $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (56)

  • Theorem 1.1: Theorem \ref{['T:WF']}
  • Proposition 2.1: kaplan2022classification*Section 4, zou2022metaplectic*Theorem 2.1
  • Remark 2.2
  • Lemma 2.3
  • Corollary 2.4
  • proof
  • Remark 2.5
  • Proposition 2.6
  • proof
  • Lemma 2.7
  • ...and 46 more