Eigenvalues of Hecke operators on Hilbert modular groups

Abstract

We consider cuspidal representations in spaces of automorphic forms for the congruence subgroup $\Gamma_0(I)$ of Hilbert modular groups for some number field $F$. To each such representation are associated the eigenvalue $\lambda_j$ of the Casimir operator at each real place $j$ of $F$, and the number $\ld_{\mathfrak p}$ parametrizing the eigenvalue of the Hecke operator $T_{\mathfrak p^2}$ at each finite place $\mathfrak p$ outside the ideal $I$. We study the joint distribution of the $\lambda_j$ for all real places $j$, and the $\ld_{\mathfrak p}$ for finitely many $\mathfrak p$ outside $I$, over the cuspidal representations. This distribution is given by the product of the Plancherel measure at each real place and the Sato-Tate measure at each finite place.