Joint distribution of eigenvalues of Hecke and Casimir operators for Hilbert Maass forms

Abstract

Let $F$ be a totally real number field, $\mathcal{O}_{F}$ the ring of integers, $\mathfrak a$ and $\mathfrak I$ integral ideals and let $\chi$ a character of $\mathbb{A}_F^\times/F^\times$. For each prime ideal $\mathfrak{p}$ in $\mathcal{O}_{F}$, $\mathfrak{p}\nmid \mathfrak{I}$ let $T_{\mathfrak{p}}$ be the Hecke operator acting on the space of Maass cusp forms on $L^2(\mathrm{GL}_{2}(F) \backslash \mathrm{GL}_{2}(\mathbb{A}_F))$. In this paper we investigate the distribution of joint eigenvalues of the Hecke operators $T_{\mathfrak{p}}$ and of the Casimir operators $C_{j}$ in each archimedean component of $F$, for $1\le j \le d$. Summarily, we prove that given a family of expanding compact subsets $\Omega_{t}$ of $\mathbb{R}^{d}$ as $t \rightarrow \infty$, and an interval $I_{\mathfrak{p}} \subseteq [-2,2]$, then, if $\mathfrak{p} \nmid \mathfrak{I}$ is a square in the narrow class group of $F$, there are infinitely many automorphic forms having eigenvalues of $T_{\mathfrak{p}}$ in $I_{\mathfrak{p}}$, distributed on $I_{\mathfrak{p}}$ according to a polynomial multiple of the Sato-Tate measure and having their Casimir eigenvalues in the region $\Omega_{t}$, distributed according to the Plancherel measure.