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Gelfand--Graev representation as a Hecke algebra module of simple types of a finite central cover of $\mathrm{GL}(r)$

Jiandi Zou

TL;DR

We address the problem of determining Whittaker (Gelfand--Graev) dimensions for genuine representations of tame Kazhdan--Patterson or Savin covers of $G=\mathrm{GL}_r(F)$. The authors realize the $Gelfand--Graev$ representation as a simple-type Hecke algebra module, decompose it into pieces indexed by $\mathcal{X}(\lambda)/\mathfrak{S}_k$, and give an explicit formula for each piece. They deduce that the Whittaker dimension for a discrete series representation with simple type $(\overline{J},\lambda)$ equals $|\mathcal{X}(\lambda)/\mathfrak{S}_k|$, and by Zelevinsky classification extend this to all irreducibles. The proof employs a three-step strategy blending cuspidal reduction, transfer from finite general linear groups, Jacquet modules, and Whittaker-uniqueness results of Paskunas--Stevens, tailored to the KP and Savin covers. The results have implications for metaplectic Langlands-type correspondences and applications in the doubling method and related conjectures.

Abstract

For an $n$-fold Kazhdan--Patterson cover or Savin's cover of a general linear group over a non-archimedean local field of residual characteristic $p$ with $\mathrm{gcd}(n,p)=1$, we realize the Gelfand--Graev representation as a Hecke algebra module of a simple type and study its explicit expression. As a main corollary, we calculate the Whittaker dimension of every discrete series representation of such a cover. Using Zelevinsky's classification, this theoretically gives the Whittaker dimension of every irreducible representation.

Gelfand--Graev representation as a Hecke algebra module of simple types of a finite central cover of $\mathrm{GL}(r)$

TL;DR

We address the problem of determining Whittaker (Gelfand--Graev) dimensions for genuine representations of tame Kazhdan--Patterson or Savin covers of . The authors realize the representation as a simple-type Hecke algebra module, decompose it into pieces indexed by , and give an explicit formula for each piece. They deduce that the Whittaker dimension for a discrete series representation with simple type equals , and by Zelevinsky classification extend this to all irreducibles. The proof employs a three-step strategy blending cuspidal reduction, transfer from finite general linear groups, Jacquet modules, and Whittaker-uniqueness results of Paskunas--Stevens, tailored to the KP and Savin covers. The results have implications for metaplectic Langlands-type correspondences and applications in the doubling method and related conjectures.

Abstract

For an -fold Kazhdan--Patterson cover or Savin's cover of a general linear group over a non-archimedean local field of residual characteristic with , we realize the Gelfand--Graev representation as a Hecke algebra module of a simple type and study its explicit expression. As a main corollary, we calculate the Whittaker dimension of every discrete series representation of such a cover. Using Zelevinsky's classification, this theoretically gives the Whittaker dimension of every irreducible representation.

Paper Structure

This paper contains 26 sections, 32 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main theorem and applications
  4. Sketch of proof
  5. Acknowledgement
  6. Notations
  7. Metaplectic covers of $\operatorname{GL}_{r}(F)$
  8. General theory for a finite central cover
  9. Metaplectic covers of $\operatorname{GL}_{r}(F)$
  10. KP-covers and S-cover
  11. Simple type theory
  12. Strata, groups and simple characters
  13. Simple types
  14. Hecke algebras
  15. Main theorem
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $\overline{G}$ be either a tame Kazhdan--Patterson cover or the Savin cover of $G=\operatorname{GL}_r(F)$. Then we have a decomposition of $\mathcal{H}$-modules such that each $\mathcal{V}_{\mathcal{O}}^{\lambda}$ is isomorphic to $\mathcal{A}\otimes_{\mathbb{C}}(\mathcal{H}_{0}\otimes_{\mathcal{H}_{\mathcal{O}}}\varepsilon_{\mathcal{O}})$, where

Theorems & Definitions (56)

  • Theorem 1.1: cf. Theorem \ref{['thmmain']}
  • Corollary 1.2: cf. Proposition \ref{['propdiscreteWhittaker']}
  • Remark 3.1
  • Proposition 4.1: zou2023simple*§ 6.2, § 6.3
  • Proposition 4.2: zou2023simple*Theorem 6.15
  • Example 4.3
  • Theorem 4.4: zou2023simple*§ 3.4, Theorem 6.15, Theorem 6.16, § 7.4
  • Theorem 5.1
  • Remark 5.2
  • Remark 5.3
  • ...and 46 more