Limit formulas of period integrals for a certain symmetric pair

Let G be the unitary group of a non-degenerate Hermitian space and H the stabilizer of a one-dimensional positive definite subspace of the Hermitian space. For a uniform lattice Γ in G such that Γ∩H is a uniform lattice of H, we introduce the (averaged) H-period integrals of automorphic forms on Γ\G; we study their behavior as Γ shrinks to the identity along a tower of lattices in G and prove a limit formula of the H-period integrals.