Periods of automorphic forms: the case of \((\mathrm{U}_{n+1}\times \mathrm{U}_{n},\mathrm{U}_{n})\) (Q1755436)

In this paper the authors compute regularised period integrals of automorphic forms associated to unitary groups. More precisely, they compute the regularised periods of certain cuspidal Eisenstein series as well as their residues. As an application, the authors make further progress on the Gan-Gross-Prasad conjecture for unitary groups. Recall that this conjecture gives a representation theoretic criterion for non-vanishing of the central critical value of a certain Rankin-Selberg automorphic \(L\)-function [\textit{W. T. Gan} et al., Astérisque 346, 1--109 (2012; Zbl 1280.22019)]. The original Gan-Gross-Prasad conjecture for unitary groups as well as a more refined version has been proved by \textit{W. Zhang} [Ann. Math. (2) 180, No. 3, 971--1049 (2014; Zbl 1322.11048); J. Am. Math. Soc. 27, No. 2, 541--612 (2014; Zbl 1294.11069)] using the relative trace formula of \textit{H. Jacquet} and \textit{S. Rallis} [Clay Math. Proc. 13, 205--264 (2011; Zbl 1222.22018)], under certain local conditions on the intervening representations. Using their period computations, the authors are able to relax these local conditions (see Theorem 5.1 for the precise statement).