On a certain local identity for Lapid-Mao's conjecture and formal degree conjecture : even unitary group case

Lapid and Mao formulated a conjecture on an explicit formula of Whittaker Fourier coefficients of automorphic forms on quasi-split classical groups and metaplectic groups as an analogue of Ichino-Ikeda conjecture. They also showed that this conjecture is reduced to a certain local identity in the case of unitary groups. In this paper, we study even unitary group case. Indeed, we prove this local identity over $p$-adic fields. Further, we prove an equivalence between this local identity and a refined formal degree conjecture over any local field of characteristic zero. As a consequence, we prove a refined formal degree conjecture over $p$-adic fields and we get an explicit formula of Whittaker Fourier coefficients under certain assumptions.