An asymptotic formula of spectral average of central $L$-values on $\mathbf{G}\mathbf{S}\mathbf{p}(2)$ for square free levels

We develop a new kind of relative trace formulas on ${\mathbf{P}\mathbf{G}\mathbf{S}\mathbf{p}}_2$ involving the Bessel periods and the Rankin-Selberg type integral a la Piatetski-Shapiro for Siegel cusp forms on its spectral side. As an application, a version of weighted equidistribution theorems for the Satake parameters of Siegel cusp forms of square-free level and of scalar weights is proved.