Impact of Eigenspaces on Hyperbolic Spaces and Formal Dimension with Limit of Period Integrals on Symmetric Spaces

Abstract

We give a complete description of all closed G- invariant subspace of the eigenspace. The eigenspace representation are compare with principal series representation , using Poisson- transformation. We show and compute the formal dimension for semisimple symmetric spaces. We introduce the stabilizer of one - dimension positive definite subspace of the Hermition period integrals of outomorphic forms on the quotient of informal lattice and the unitary group of anon-degenerate Hermition space.