Weyl's Law for Arbitrary Archimedean Type

TL;DR

This paper extends Weyl's Law for cusp forms from the spherical spectrum to arbitrary Archimedean types, providing an asymptotic count of cusp forms with general Archimedean conditions using advanced harmonic analysis tools.

Contribution

It generalizes Weyl's Law for cusp forms to all Archimedean types, replacing the spherical case methods with Arthur's Paley-Wiener theorem and multipliers.

Findings

Weyl's Law holds for cusp forms of arbitrary Archimedean type.

Asymptotic formula involves the dimension of the Archimedean type.

Method extends previous spherical spectrum results to more general cases.

Abstract

We generalize the work of Lindenstrauss and Venkatesh establishing Weyl's Law for cusp forms from the spherical spectrum to arbitrary Archimedean type. Weyl's law for the spherical spectrum gives an asymptotic formula for the number of cusp forms that are bi-spherical in terms of eigenvalue T of the Laplacian. We prove an analogous asymptotic holds for cusp forms with Archimedean type {\tau}, where the main term is multiplied by dim {\tau}. While in the spherical case the surjectivity of the Satake Map was used, in the more general case that is not available and we use Arthur's Paley-Wiener theorem and multipliers.