On a pairing of Goldberg-Shahidi for even orthogonal groups

Let σ\otimesπ be a supercuspidal representation of SO(2n) \times GL(2n) over a p-adic field with π 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 σ and π 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 π does not come from SO(2n+1).

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