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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?

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    $\begingroup$ This is a very interesting question, and there is a lot to be said about it. Here are just a few remarks: 1. The role of eigenvalues is played by local systems, not opers. 2. According to wikipedia, the general statement of Ramanujan-Petersson is that for a globally generic cuspidal automorphic representation, each local component is tempered. Globally generic is irrelevant in the geometric setting, and temperedness corresponds (on the LocSys) side to lying in QCoh(LocSys) rather than IndCoh. So if you assume the statement of geometric Langlands (D-mod(Bun_G) is equivalent to IndCoh_N(LocSys)) $\endgroup$ – dhy Jan 30 at 18:37
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    $\begingroup$ then Ramanujan-Petersson should be the statement that the image of D-mod_cusp(Bun_G) under this equivalence should be contained in QCoh(LocSys). 3. However, you would want a purely automorphic (without invoking Langlands) statement, in terms of the Hecke action. For a purely automorphic formulation of temperedness, see section 12.8 of the Arinkin-Gaitsgory singular support paper. So now the question is if D-mod_cusp(Bun_G) is contained in D-mod_temp^x(Bun_G) for all points $x$ of your curve. Anyways, this is getting pretty technical... I can try to clarify individual points if asked. $\endgroup$ – dhy Jan 30 at 18:48
  • $\begingroup$ @dhy, much thanks for your reply! Can I in particular request elaboration of the following point: Suppose I take an object in D-mod_temp^x(Bun_G) then what is the relation between the singular support of this object and the action of geometric Hecke operators? Of course elaboration on any other points is also appreciated! $\endgroup$ – shehryar sikander Jan 30 at 20:44
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    $\begingroup$ Unfortunately I'm not sure if I can phrase the relation without introducing a lot of notation. This is the subject of section 12.8 of Arinkin-Gaitsgory. An important subtlety here to be aware of is that the true category of Hecke operators is larger than the naive one described by Mirkovic-Vilonen and is only visible at the derived level. $\endgroup$ – dhy Jan 31 at 5:11
  • $\begingroup$ Thanks for the insight! And for the reference, i'm looking at it now. I was hoping that the Ramanujan-Peterrson property for the eigen \check{G}-Local System would be intrinsic, like a bound on the eigen values of the monodromy of the local system. Perhaps one should take inspiration from the Ramanujan-Petersson for automorphic forms over function field of a curve over a finite field; which is more similar to Geometric Langlands. $\endgroup$ – shehryar sikander Jan 31 at 23:04

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