A local Weyl’s law, the angular distribution and multiplicity of cusp forms on product spaces

Abstract:

Let $\Gamma /\mathcal {H}$ be a finite volume symmetric space with $\mathcal {H}$ the product of half planes. Let ${\Delta _i}$ be the Laplacian on the $i$th half plane, and assume that we have a cusp form $\phi$, so we have ${\Delta _i}\phi = {\lambda _i}\phi$ for $i = 1,2, \ldots ,n$. Let $\vec \lambda = ({\lambda _1}, \ldots ,{\lambda _n})$ and let \[ R = \sqrt {r_1^2 + \cdots + r_n^2} \] with $r_i^2 + \frac {1} {4} = {\lambda _i}$. Letting $\vec r = ({r_1}, \ldots ,{r_n})$, we let $M(\vec r)$ denote the dimension of the space of cusp forms with eigenvalue $\vec \lambda$. More generally, let $M(\vec r,a)$ denote the number of independent eigenfunctions such that the $\vec r$ associated to an eigenfunction is inside the ball of radius $a$, centered at $\vec r$. We will define a function $f(\vec r)$, which is generally equal to a linear sum of products of the ${r_i}$. We prove the following theorems. Theorem 1. \[ M(\vec r) = O\left (\frac {f(\vec r)} {(\log R)^n} \right ). \] Theorem 2. \[ M (\vec {r}, A) = 2^n f(\vec {r})+O\left (\frac {f(\vec r)}{\log R} \right ). \]