The guiding Sato–Tate principle for families of automorphic representations

Let $F$ be a number field, let $G$ be a reductive group, and let $\{\mathcal F_k\}_{k\geqslant1}$ be a “general” sequence of finite families of automorphic representations of $G(\mathbb A_F)$ such that $|\mathcal F_k|\longrightarrow\infty$ as $k\longrightarrow\infty$. For $\theta\in{\mathscr C}(\Gamma_1)$, let ${\mathcal V}_F(\theta)$ be the corresponding set of finite places and let $\widehat{\mu}^{\mathrm{ST}}_\theta$ be the Sato–Tate measure. Sato–Tate principle for families. The local components $\{\pi_v\in\mathcal F_k\}$ should be equidistributed according to $\widehat{\mu}^{\mathrm{ST}}_\theta$ as both $k$ and $v$ tend to infinity, subject to $v\in{\mathcal V}_F(\theta)$ and every member of $\mathcal F_k$ being unramified at $v$. The source explicitly presents this as a guiding principle rather than a rigorous conjecture; consequently its status as a formal conjecture is uncertain.