Expression of an epsilon factor of pairs by an integral formula (Q2922902)

Let \(E/F\) be a quadratic extension of non-archimedean local fields of characteristic \(0\) and let \(\pi\) and \(\sigma\) be two tempered irreducible smooth representations of GL\((d,E)\) and GL\((m,E)\), where \(d\) and \(m\) are nonnegative integers of distinct parities. Assume moreover that \(\pi\) and \(\sigma\) are conjugate-dual, that is \(\pi^c\simeq \check{\pi}\), \(\sigma^c\simeq \check{\sigma}\), where \(c\) denotes the action of the non-trivial \(F\)-automorphism of \(E\), and a symbol \(\check{}\) denotes a contragredient representation. In particular \(\pi\) and \(\sigma\) may be extended to unitary representations \(\tilde{\pi}\) and \(\tilde{\sigma}\) of GL\((d,E)\ltimes \{ 1,c\}\) and GL\((m,E)\ltimes \{ 1,c\}\) respectively. Finally let \(\varepsilon (1/2 ,\pi\times\sigma ,\psi_E )\) be the epsilon factor of pair, for a fixed non-trivial additive character \(\psi_E\) of \(E\), as defined in [\textit{H. Jacquet} et al., Am. J. Math. 105, 367--464 (1983; Zbl 0525.22018)].NEWLINENEWLINEThe author proves an integral formula for \(\varepsilon (1/2 ,\pi\times\sigma ,\psi_E )\) involving the Harish-Chandra characters of \(\check{\pi}\) and \(\check{\sigma}\). He hopes that one should be able to apply this formula to obtain a proof of the local Gan-Gross-Prasad conjecture for unitary groups, cf. conjecture 17.3 of [\textit{W. T. Gan} et al., Sur les conjectures de Gross et Prasad. I. Paris: Société Mathématique de France (SMF) (2012; Zbl 1257.22001)]. This formula is the exact analogue of that obtained by Waldspurger for epsilon factor of pairs of auto-dual representations of general linear groups [\textit{C. Mœglin} and \textit{J.-L. Waldspurger}, Sur les conjectures de Gross et Prasad. II. Paris: Société Mathématique de France (SMF) (2012; Zbl 1257.22002)].