A weighted vertical Sato-Tate law for Maaß forms on ${\mathrm{G}\mathrm{S}\mathrm{p}}_4$

We prove a weighted Sato-Tate law for the Satake parameters of automorphic forms on ${\mathrm{G}\mathrm{S}\mathrm{p}}_4$ with respect to a fairly general congruence subgroup $H$ whose level tends to infinity. When the level is squarefree we refine our result to the cuspidal spectrum. The ingredients are the ${\mathrm{G}\mathrm{S}\mathrm{p}}_4$ Kuznetsov formula and the explicit calculation of local integrals involved in the Whittaker coefficients of ${\mathrm{G}\mathrm{S}\mathrm{p}}_4$ Eisenstein series. We also discuss how the problem of bounding the continuous spectrum in the level aspect naturally leads to some combinatorial questions involving the double cosets in $P\mathrm{\backslash }G\mathrm{/}H$, for each parabolic subgroup $P$.