Weyl’s law for the cuspidal spectrum of $\mathrm{SL}_n$ | Annals of Mathematics

Abstract Let Γ be a principal congruence subgroup of SLn(ℤ) and let σ be an irreducible unitary representation of SO(n). Let NΓcu(λ,σ) be the counting function of the eigenvalues of the Casimir operator acting in the space of cusp forms for Γ which transform under SO(n) according to σ. In this paper we prove that the counting function NΓcu(λ,σ) satisfies Weyl’s law. Especially, this implies that there exist infinitely many cusp forms for the full modular group SLn(ℤ).

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数学年刊(Annals of Mathematics)

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