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

TL;DR

This paper investigates how the endoscopic transfer for the metaplectic group Mp(2n) varies with different additive characters of a local field, providing a simple proof of the dependence of the local Langlands correspondence on these characters.

Contribution

It offers a straightforward, uniform analysis of the transfer behavior under additive character variation, avoiding complex representation theory results.

Findings

Transfer factor properties are key to understanding variation.

Provides a simple proof of the dependence of Langlands correspondence on additive characters.

Clarifies the behavior of endoscopic transfer for Mp(2n) under character changes.

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.