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Representation Theory

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On a pairing of Goldberg-Shahidi for even orthogonal groups

Author: Wen-Wei Li
Journal: Represent. Theory 17 (2013), 337-381
MSC (2010): Primary 22E50; Secondary 11F70
Published electronically: June 17, 2013
MathSciNet review: 3067291
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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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Wen-Wei Li
Affiliation: Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences 55, Zhongguancun East Road, 100190 Beijing, China

Received by editor(s): June 1, 2012
Received by editor(s) in revised form: December 6, 2012, and January 6, 2013
Published electronically: June 17, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.