In this talk, I will introduce the Sakellaridis-Venkatesh conjecture on the decomposition of global period, and give examples related to this conjecture. More specifically, the case X=SO(n-1)\SO(n) and X=U(2)\SO(5).

In both cases, I will determine the Plancherel decompositions of  L^2(X_v), where v is a local place. Then I will prove the local relative character identity.

In the global setting, I will give the factorization of the global period of X=SO(n-1)\SO(n), where the local functional comes from the local Plancherel decomposition. The example X=U(2)\SO(5) is slightly beyond the SV conjecture but we still have a decomposition of the global period as the sum of two factorizable elements.