Spectral multiplicity for $\textrm {Gl}_ n(\textbf {R})$

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 \[ N\prime (\lambda ) = {C_r}({\text {Vol}}\;X){\lambda ^{\frac {r} {2}}} + O({\lambda ^{\frac {{r - 1}} {2}}}{\lambda ^{\frac {1} {{n + 1}}}}{(\log \lambda )^{n - 1}}).\] Theorem 2. \[ {M_0}(\lambda ) = O({\lambda ^{\frac {{r - 1}} {2}}}{\lambda ^{\frac {1} {{n + 1}}}}{(\log \lambda )^{n - 1}}).\]