Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Variation of additive characters in the transfer for Mp(2n)

Wen-Wei Li

Abstract

Let $\mathrm{Mp}(2n)$ be the metaplectic group of rank $n$ over a local field $F$ of characteristic zero. In this note, we determine the behavior of endoscopic transfer for $\mathrm{Mp}(2n)$ under variation of additive characters of $F$. The arguments are based on properties of transfer factor, requiring no deeper results from representation theory. Combined with the endoscopic character relations of Luo, this provides a simple and uniform proof of a theorem of Gan-Savin, which describes how the local Langlands correspondence for $\mathrm{Mp}(2n)$ depends on the additive characters.

Variation of additive characters in the transfer for Mp(2n)

Abstract

Let be the metaplectic group of rank over a local field of characteristic zero. In this note, we determine the behavior of endoscopic transfer for under variation of additive characters of . The arguments are based on properties of transfer factor, requiring no deeper results from representation theory. Combined with the endoscopic character relations of Luo, this provides a simple and uniform proof of a theorem of Gan-Savin, which describes how the local Langlands correspondence for depends on the additive characters.

Paper Structure

This paper contains 21 sections, 17 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Main results and proofs
  4. Conventions
  5. Review of endoscopy
  6. Endoscopic data of metaplectic groups
  7. Correspondence of conjugacy classes
  8. Transfer
  9. Review of local Langlands correspondence
  10. The correspondence
  11. Endoscopic character relation
  12. A theorem of Gan--Savin
  13. Spinor norms
  14. Basic properties
  15. Effect on L-parameters
  16. ...and 6 more sections

Key Result

Theorem 1.2.1

Given $c \in F^{\times}$, define the additive character ${\uppsi}_c$ of $F$ by ${\uppsi}_c(x) = {\uppsi}(cx)$. For all bounded L-parameter $\phi$ and $\chi \in \EuScript{S}_\phi^\vee$, we have where

Theorems & Definitions (33)

  • Theorem 1.2.1: = Theorem \ref{['prop:GS']}
  • Theorem 1.2.2: = Theorem \ref{['prop:Trans-var']}
  • Lemma 3.2.1: special case of Li24a
  • Theorem 3.2.2: C. Luo Luo20
  • Definition 3.3.1
  • Theorem 3.3.2: Gan--Savin GS1
  • Lemma 4.1.1
  • proof
  • Remark 4.1.2
  • Lemma 4.1.3
  • ...and 23 more