The Ramanujan conjecture asserts that \begin{align} |\tau(p)|\leq 2p^{11/2} \end{align} where $\tau(p)$ is the $p^{th}$ Fourier coeffecient in the q-expansion of the weight 12 cusp form $\Delta(z)$. Its generalization, the Ramanujan-Petersson conjecture, asserts a bound on the eigen values of Hecke operators acting on the space of cuspidal automorphic forms.

In Geometric Langlands, the role of automorphic forms is taken up by D-Modules on $Bun_G$ and the role of Hecke operators is taken up by geometric Hecke operators which act as a correspondence from $Bun_G \to Bun_G \times X$ and act as functors from the category of D-Modules on $Bun_G$ to the category of D-Modules on $Bun_G \times X$. The eigen values of these geometric Hecke operators are $\check{G}$-Local systems on $X$ for the Langlands dual group $\check{G}$. What is the analog of Ramanujan-Petersson conjecture for the eigen values (The $\check{G}$-Local systems) of the geometric Hecke operators in this case?