Limit formulas of period integrals for a certain symmetric pair II

Abstract

Let $(G,H) = (U(p,q),U(p-1,q) \times U(1))$ and $\{\Gamma_n\}$ a tower of congruence uniform lattices in $G$. By the period integrals of automorphic forms on $\Gamma \backslash G$ along $\Gamma_n \cap H\backslash H$ , we introduce a certain discrete measure $d \mu^H_{\Gamma_n}$ on the $H$-spherical unitary dual of $G$. It is shown that the sequence of measures $d \mu^H_{\Gamma_n}$ with growing $n$ converges in a weak sense to the Plancherel measure $d \mu^H$ for the symmetric space $H\backslash G$.