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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Spectral multiplicity for $ {\rm Gl}\sb n({\bf R})$


Author: Jonathan Huntley
Journal: Trans. Amer. Math. Soc. 332 (1992), 875-888
MSC: Primary 11F72; Secondary 11F46, 11F55, 22E30, 58G25
MathSciNet review: 1102223
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Abstract: We study the behavior of the cuspidal spectrum of $ \Gamma \backslash \mathcal{H}$, where $ \mathcal{H}$ is associated to $ \operatorname{Gl}_n(R)$ and $ \Gamma $ is cofinite but not compact. By a technique that modifies the Lax-Phillips technique and uses ideas from wave equation techniques, if $ r$ is the dimension of $ \mathcal{H}$, $ {N_\alpha }(\lambda )$ is the counting function for the Laplacian attached to a Hilbert space $ {H_\alpha }$, $ {M_\alpha }(\lambda )$ is the multiplicity function, and $ {H_0}$ is the space of cusp forms, we obtain the following results:

Theorem 1. There exists a space of functions $ {H^1}$, containing all cusp forms, such that

$\displaystyle N\prime(\lambda ) = {C_r}({\text{Vol}}\;X){\lambda ^{\frac{r} {2}... ...{\frac{{r - 1}} {2}}}{\lambda ^{\frac{1} {{n + 1}}}}{(\log \lambda )^{n - 1}}).$

Theorem 2.

$\displaystyle {M_0}(\lambda ) = O({\lambda ^{\frac{{r - 1}} {2}}}{\lambda ^{\frac{1} {{n + 1}}}}{(\log \lambda )^{n - 1}}).$


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1992-1102223-5
PII: S 0002-9947(1992)1102223-5
Article copyright: © Copyright 1992 American Mathematical Society