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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Authors: Jonathan Huntley and David Tepper
Journal: Trans. Amer. Math. Soc. 330 (1992), 97-110
MSC: Primary 11F72; Secondary 11F55
MathSciNet review: 1053114
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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

$\displaystyle 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.

$\displaystyle M(\vec r) = O\left(\frac{f(\vec r)} {(\log R)^n} \right). $

Theorem 2.

$\displaystyle M (\vec{r}, A) = 2^n f(\vec{r})+O\left(\frac{f(\vec r)}{\log R} \right). $

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Article copyright: © Copyright 1992 American Mathematical Society

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