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.
Categories: math.NT