On a pairing of Goldberg-Shahidi for even orthogonal groups

TL;DR

This paper investigates a pairing between supercuspidal representations of even orthogonal and general linear groups, aiming to connect it with endoscopic transfer and verify predictions using Arthur's classification.

Contribution

It extends Goldberg-Shahidi's pairing analysis to general n and verifies related predictions under specific assumptions, utilizing Arthur's endoscopic classification.

Findings

Verified some predictions of Goldberg-Shahidi pairing for general n

Connected the pairing's elliptic part to twisted endoscopic transfer

Built upon Arthur's classification and Spallone's improvements

Abstract

Let {\sigma}\otimes{\pi} be a supercuspidal representation of SO(2n) \times GL(2n) over a p-adic field with {\pi} selfdual, where SO(2n) stands for a quasisplit even special orthogonal group. In order to study its normalized parabolic induction to SO(6n), Goldberg and Shahidi defined a pairing R between the matrix coefficients of {\sigma} and {\pi} which controls the residue of the standard intertwining operator. The elliptic part of R is conjectured to be related to twisted endoscopic transfer. Based on Arthur's endoscopic classification and Spallone's improvement of Goldberg-Shahidi program, we will verify some of their predictions for general n, under the assumption that {\pi} does not come from SO(2n+1).