On a pairing of Goldberg-Shahidi for even orthogonal groups

Li, Wen-Wei

doi:10.1090/S1088-4165-2013-00435-1

On a pairing of Goldberg-Shahidi for even orthogonal groups
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by Wen-Wei Li

Represent. Theory 17 (2013), 337-381

DOI: https://doi.org/10.1090/S1088-4165-2013-00435-1

Published electronically: June 17, 2013

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Abstract:

Let $\pi \otimes \sigma$ be a supercuspidal representation of $\mathrm {GL}(2n) \times \mathrm {SO}(2n)$ over a $p$-adic field with $\pi$ selfdual, where $\mathrm {SO}(2n)$ stands for a quasisplit even special orthogonal group. In order to study its normalized parabolic induction to $\mathrm {SO}(6n)$, Goldberg and Shahidi defined a pairing $R$ between the matrix coefficients of $\pi$ and $\sigma$ which controls the residue of the standard intertwining operator. The elliptic part $R_\text {ell}$ 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 $\mathrm {SO}(2n+1)$.

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