A Local Weyl's Law, the Angular Distribution and Multiplicity of Cusp Forms on Product Spaces

Let Y\ßf be a finite volume symmetric space with %? the product of half planes. Let A, be the Laplacian on the ¡th half plane, and assume that we have a cusp form , so we have A, = X¡ for i = 1, 2, ... , n . Let A = (Ai,..., An) and let R = \/>-\ + ■ ■ ■ + r2„ with rj + j = A, . Letting r = (ri.rn), we let M(r) denote the dimension of the space of cusp forms with eigenvalue A. More generally, let M(r, a) denote the number of independent eigenfunctions such that the r associated to an eigenfunction is inside the ball of radius a , centered at f. We will define a function f(r), which is generally equal to a linear sum of products of the r¡. We prove the following theorems. Theorem 1. \(\ogR)»J Theorem 2. M(r,A) = 2"f(r) + o(^L).