On Splitting Invariants and Sign Conventions in Endoscopic Transfer

TL;DR

This paper introduces a twisted version of the Langlands-Shelstad splitting invariant to improve endoscopic transfer factors and corrects a sign error to ensure compatibility with the Langlands correspondence.

Contribution

It proposes a new twisted splitting invariant and defines an improved transfer factor, correcting previous sign errors for better compatibility with Langlands correspondence.

Findings

Introduces a twisted splitting invariant for endoscopic transfer.

Defines a new twisted transfer factor $ ext{Delta}_I$.

Corrects a sign error in twisted transfer definitions.

Abstract

The transfer factors for standard endoscopy involve, among other things, the Langlands-Shelstad splitting invariant. This note introduces a twisted version of that splitting invariant. The twisted splitting invariant is then used to define a better twisted factor $\Delta_I$. In addition we correct a sign error in the definition of twisted transfers. There are two ways to correct the sign error. One way yields twisted transfer factors $\Delta'$ that are compatible with the classical Langlands correspondence. The other way yields twisted transfer factors $\Delta_D$ that are compatible with a renormalized version of the Langlands correspondence.