Spectral multiplicity for 𝐺𝑙_{𝑛}(𝑅)

Abstract

We study the behavior of the cuspidal spectrum of Γ ∖ H \Gamma \backslash \mathcal {H} , where H \mathcal {H} is associated to Gl n ⁡ ( R ) \operatorname {Gl}_n(R) and Γ \Gamma is cofinite but not compact. By a technique that modifies the Lax-Phillips technique and uses ideas from wave equation techniques, if r r is the dimension of H \mathcal {H} , N α ( λ ) {N_\alpha }(\lambda ) is the counting function for the Laplacian attached to a Hilbert space H α {H_\alpha } , M α ( λ ) {M_\alpha }(\lambda ) is the multiplicity function, and H 0 {H_0} is the space of cusp forms, we obtain the following results: Theorem 1. There exists a space of functions H 1 {H^1} , containing all cusp forms, such that \[ N ′ ( λ ) = C r ( Vol X ) λ r 2 + O ( λ r − 1 2 λ 1 n + 1 ( log ⁡ λ ) n − 1 ) . N\prime (\lambda ) = {C_r}({\text {Vol}}\;X){\lambda ^{\frac {r} {2}}} + O({\lambda ^{\frac {{r - 1}} {2}}}{\lambda ^{\frac {1} {{n + 1}}}}{(\log \lambda )^{n - 1}}). \] Theorem 2. \[ M 0 ( λ ) = O ( λ r − 1 2 λ 1 n + 1 ( log ⁡ λ ) n − 1 ) . {M_0}(\lambda ) = O({\lambda ^{\frac {{r - 1}} {2}}}{\lambda ^{\frac {1} {{n + 1}}}}{(\log \lambda )^{n - 1}}). \]