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The local converse theorem for $Mp_{2n}$ : the generic case

Jaeho Haan

TL;DR

This work establishes a local converse theorem for quasi-split metaplectic groups $\widetilde{\mathrm{Sp}}_{2n}$ by leveraging a precise local theta correspondence with $\mathrm{SO}_{2n+1}$ over local fields of characteristic not equal to $2$. The authors show that genericity and local $\gamma$-factors are preserved under the theta correspondence, enabling transfer of the LCT from odd orthogonal groups to metaplectic groups and proving stability of local gamma-factors in this setting, including the positive characteristic case under a generic-factor hypothesis. A global rigidity theorem for irreducible generic cuspidal automorphic representations of $\widetilde{\mathrm{Sp}}_{2n}$ is also proved via global theta lifts and Kudla-type global-to-local arguments. The approach is theta-correspondence–driven and provides an independent route to the local Langlands–type classification for metaplectic groups, avoiding reliance on Arthur’s framework and highlighting the utility of theta lifts for both local and global problems.

Abstract

In this paper, we establish the local converse theorem and the stability of local gamma factors for $\Mp_{2n}$ via the precise local theta correspondence between $\Mp_{2n}$ and $\SO_{2n+1}$ over local fields of characteristic not equal to 2. We also prove the rigidity theorem for irreducible generic cuspidal automorphic representations of $\Mp_{2n}$ over number fields.

The local converse theorem for $Mp_{2n}$ : the generic case

TL;DR

This work establishes a local converse theorem for quasi-split metaplectic groups by leveraging a precise local theta correspondence with over local fields of characteristic not equal to . The authors show that genericity and local -factors are preserved under the theta correspondence, enabling transfer of the LCT from odd orthogonal groups to metaplectic groups and proving stability of local gamma-factors in this setting, including the positive characteristic case under a generic-factor hypothesis. A global rigidity theorem for irreducible generic cuspidal automorphic representations of is also proved via global theta lifts and Kudla-type global-to-local arguments. The approach is theta-correspondence–driven and provides an independent route to the local Langlands–type classification for metaplectic groups, avoiding reliance on Arthur’s framework and highlighting the utility of theta lifts for both local and global problems.

Abstract

In this paper, we establish the local converse theorem and the stability of local gamma factors for via the precise local theta correspondence between and over local fields of characteristic not equal to 2. We also prove the rigidity theorem for irreducible generic cuspidal automorphic representations of over number fields.
Paper Structure (15 sections, 20 theorems, 86 equations)

This paper contains 15 sections, 20 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Preliminaries
  4. Metaplectic and orthogonal groups
  5. Symplectic groups
  6. Metaplectic groups
  7. Orthogonal groups
  8. Genuine representations
  9. Generic characters and generic representations
  10. $\gamma$-factors
  11. Reduction of Theorem \ref{['main']} to $\lambda=1$ cases
  12. Local theta correspondence
  13. Computation of the twisted Jacquet module of the Weil representation
  14. The proof
  15. The rigidity theorem for $\widetilde{\mathrm{Sp}}_{2n}(\mathbb{A})$

Key Result

Theorem 1.1

Assume the above working hypothesis on $\operatorname{char}(F)=p$ case. Let $\widetilde{\pi}$ and $\widetilde{\pi}'$ be irreducible admissible $\mu_{\lambda}'$-generic representations of $\widetilde{\mathrm{Sp}}_{2n}(F)$, satisfying the condition holds for all irreducible supercuspidal representation $\sigma$ of $\mathrm{GL}_i(F)$ with $1\le i \le n$. Then, it follows that

Theorems & Definitions (37)

  • Theorem 1.1: Local Converse Theorem for $\widetilde{\mathrm{Sp}}_{2n}$
  • Theorem 1.2: Stability of local $\gamma$-factors for $\widetilde{\mathrm{Sp}}_{2n}$
  • Theorem 1.3: Rigidity theorem for $\widetilde{\mathrm{Sp}}_{2n}$
  • Proposition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4: Local Converse Theorem for $\widetilde{\mathrm{Sp}}_{2n}$
  • Theorem 3.1: Howe duality
  • Theorem 3.2: GI1
  • ...and 27 more