In this talk, we will present our recent results on the local multiplicity formula of strongly tempered spherical subgroups. In particular, we formulate the epsilon dichotomy conjecture for those spherical subgroups and prove this conjecture for the tempered representations of endoscopy type. Globally, the corresponding automorphic periods are related to the central values of certain L-functions of symplectic type. The conjectural Ikeda–Ichino type formulas are given. This is a joint project with Chen Wan at Rutgers University, Newark.