Weyl’s law for arbitrary archimedean type

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-\(K_{\infty }\) invariant in terms of eigenvalue T of the Laplacian. We prove that 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.