To motivate this question, I'm going to try and explain some background notions. This won't be absolutely necessary for experts, but I want to be vaguely honest about where this question comes from. I also want to say that this is the type of question that I would ask at a coffee break during a conference about this subject. If my question is too disorganized for mathoverflow, I apologize, and there will be no hard feelings if this question is closed because it is too speculative.
Given a manifold $M$, one has a canonical symplectic structure on the cotangent bundle $T^{*}M$. Given a Poisson sub-algebra of functions $A$ on $T^{*}M$, one aim of quantization asks if there exists an algebra of differential operators $A^{'}$ on $M,$ (generally they may be differential operators on sections of line bundles over $M$), for which the principal symbol map $A^{'}\rightarrow A$ is a morphism, where commutator of differential operators corresponds to Poisson bracket of functions on $T^{*}M.$
This question, which has its roots in the passage from classical to quantum mechanics, was employed by Beilinson-Bernstein in their localization procedure for producing modules over the universal enveloping algebra of a semisimple complex Lie algebra $\mathfrak{g}$ from $D$-modules on the flag variety $G/B$ where $G$ integrates $\mathfrak{g}$ and $B<G$ is a Borel sub-group.
Beilinson and Drinfeld, in a stroke of genius, realized that for $G$ a connected, complex reductive algebraic group over $\mathbb{C},$ this process can be applied to $T^{*}{Bun}_{G}(X)$ where $X$ is a compact Riemann surface, and ${Bun}_{G}(X)$ is the moduli stack of principal $G$-bundles over $X.$ Here, the sub-algebra of functions they consider is the Hitchin integral system. Morally speaking, a cotangent vector $\xi$ to a principal $G$ bundle $E_{G}$ is a global section $\xi\in H^{0}(X, K\otimes E_{G}(\mathfrak{g})),$ and a $G$-invariant symmetric homogeneous polynomial on the Lie algebra $\mathfrak{g}$ (almost) produces an function on $T^{*}{Bun}_{G}(X)$ via evaluation: this quickly leads to a completely integrable system on $T^{*}{Bun}_{G}(X)$ called the Hitchin fibration.
What Beilinson and Drinfeld do is produce a commuting algebra of (twisted) differential operators on $Bun_{G}(X)$ whose principal symbols map to functions appearing in the Hitchin integrable system.
While certainly an attractive story on face, as a pedestrian differential geometer, the theory of stacks and local algebra through which this construction passes hides, for me, what these differential operators actually are.
This is where my question begins. Instead of the stack $Bun_{G}(X),$ consider instead the projective variety (analytic space) parameterizing stable $G$-bundles on $X.$ I'm being very loose here, so maybe I want to switch to $GL(n, \mathbb{C})$, and also add extra hypotheses to make the following question well formed.
Question: Is there an avatar of the Beilinson-Drinfeld commuting algebra of differential operators on the moduli space of stable $G$-bundles on $X?$ For $GL(1, \mathbb{C})$, I think this question has a nice answer, and is basically part of the passage from abelian class field theory to geometric class field theory as espoused by many important figures in the study of the geometric Langlands conjecture.
Moving to non-abelian groups like $GL(2,\mathbb{C}),$ I already have no precise idea what might be going on. In this case, line bundles over the moduli space of stable $G$-bundles are a very important object, and subsequent objects like generalized Theta functions play an important role, for example in the Verlinde formula. It's possible that hidden inside these ideas I should find the differential operators I am seeking, but this is the point of my question.
Refined question: Is there a commuting family of differential operators on the space of stable $G$-bundles which corresponds to the Beilinson-Drinfeld construction, which has a formulation in terms of generalized theta functions etc.
In an attempt to not be a total idiot, I understand that in the Beilinson-Drinfeld application, they're focusing on $G$-bundles which are very unstable, those corresponding to $G$-opers, and therefore the stack point of view is essential. In this vein, I'm not asking for an explanation of their work which can be understood in the language of stable bundles. I'm just asking if their construction produces something interesting, perhaps already studied before, when restricted to the space of stable bundles.
I apologize if this is a series of paragraphs, each compounding the next, resulting in a question that makes no sense. If this is not the case, I appreciate any responses, and thank you for reading to the end.
