On a pairing of Goldberg-Shahidi for even orthogonal groups (Q2836488)

In [Duke Math. J. 92, No. 2, 255--294 (1998; Zbl 0938.22014)], \textit{D. Goldberg} and \textit{F. Shahidi} defined and studied a pairing \(R\) between the matrix coefficients of a self-dual supercuspidal representation \(\pi\) of \(\mathrm{GL}(2n, F)\) and a supercuspidal representation \(\sigma\) of \(\mathrm{SO}(2n, F)\) (with \(F\) a non-Archimedean local field), which controls the residue at \(0\) of the standard intertwining operator of the representation \(I_P(\pi\otimes \sigma)\) of \(\mathrm{SO}(6n)\), parabolically induced from the maximal parabolic \(P\) whose Levi subgroup is isomorphic to \(\mathrm{GL}(2n)\times\mathrm{ SO}(2n)\). This pairing is composed of the two terms -- elliptic \(R_{\mathrm{ell}}\) and singular \(R_{\mathrm{sing}}\), whose respective non-vanishing is conjecturally related to the poles at \(s=0\) of the Rankin-Selberg \(L\)-function \(L(s, \pi\times\sigma)\) and the exterior square \(L\)-function \(L(s, \pi, \Lambda^2)\), as explained in [\textit{F. Shahidi}, Int. Math. Res. Not. 2008, Article ID rnn095, 13 p. (2008; Zbl 1232.11094)]. F. Shahidi also conjectured the relationship between non-vanishing of \(R_{\mathrm{ell}}\) and twisted endoscopy; this conjectural relationship was proved in the case \(n=1\) by \textit{F. Shahidi} and \textit{S. Spallone} [Compos. Math. 146, No. 3, 772--794 (2010; Zbl 1193.22019)]; the proof relied on the character identities of Labesse-Langlands.NEWLINENEWLINEThe present article establishes the relationship between non-vanishing of \(R_{\mathrm{ell}}\) and twisted endoscopy for general \(n\), under the hypothesis that \(\pi\) does not come by endoscopic transfer from \(\mathrm{SO}(2n+1)\). The proof relies on J. Arthur's endoscopic classification of representations of orthogonal groups, [\textit{J. Arthur}, The endoscopic classification of representations. Orthogonal and symplectic groups. Providence, RI: American Mathematical Society (AMS) (2013; Zbl 1310.22014)].