B. H. Gross and D. Prasad [Can. J. Math. 44, No. 5, 974–1002 (1992; Zbl 0787.22018)] conjectured that whether the restriction of a cuspidal representation of an orthogonal group contains a cuspidal representation of an orthogonal group of smaller rank is determined by the nonvanishing of certain special value of \(L\)-function. Later A. Ichino and T. Ikeda [Geom. Funct. Anal. 19, No. 5, 1378–1425 (2010; Zbl 1216.11057)] gave an explication of their conjecture – they relates the special value to a period integral. This paper gives the analogue of Ichino-Ikeda’s work in the setting of unitary groups.
The main result is a conjectured explicit formula between a period integral on unitary group and special value of AN \(L\)-function. To state the conjecture, the author needs to check the convergence of local integrals and calculate the integrals in the unramified setting. The author also proves the conjecture in the low rank cases of \(\mathrm{U}(2)\times \mathrm{U}(1)\) and \(\mathrm{U}(3)\times \mathrm{U}(2)\). The first is based on J. L. Waldspurger’s work on toric periods [Compos. Math. 54, 173–242 (1985; Zbl 0567.10021)], the later is based on theta correspondence and Siegel-Weil formula.