On the Gross-Prasad conjecture with its refinement for $\left(\mathrm{SO}\left(5\right),\mathrm{SO}\left(2\right)\right)$ and the generalized B\"ocherer conjecture

We investigate the Gross-Prasad conjecture and its refinement for the Bessel periods in the case of $\left(\mathrm{SO}\left(5\right),\mathrm{SO}\left(2\right)\right)$. In particular, by combining several theta correspondences, we prove the Ichino-Ikeda type formula for any tempered irreducible cuspidal automorphic representations. As a corollary of our formula, we prove an explicit formula relating certain weighted averages of Fourier coefficients of holomorphic Siegel cusp forms of degree two which are Hecke eigenforms to central special values of $L$-functions. The formula is regarded as a natural generalization of the B\"ocherer conjecture to the non-trivial toroidal character case.