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Frenkel-Gross’ irregular connection and Heinloth-Ngô-Yun’s are the same. (English) Zbl 1393.14011
The goal of this paper is to compare two connections on $$\mathbb{G}_m$$, one constructed by E. Frenkel and B. Gross [Ann. Math. (2) 170, No. 3, 1469–1512 (2009; Zbl 1209.14017)], and the other by J. Heinloth et al. [Ann. Math. (2) 177, No. 1, 241–310 (2013; Zbl 1272.14012)]. These connections have regular singularity at $$0 \in \mathbb{P}^1$$, with principal unipotent monodromy; at the point $$\infty$$ they are irregular.
The idea is to use the automorphic to Galois direction in the geometric Langlands correspondence for $$D$$-modules. The machinery of Beilinson-Drinfeld produces an automorphic $$D$$-module $$\mathrm{Aut}_{\mathcal{E}}$$ on $$\mathrm{Bun}_{\mathcal{G}}$$ from the Heinloth-Ngo-Yun connection, where $$\mathcal{G}$$ is an appropriate model of the reductive group $$G$$ in question, with prescribed ramification at $$\{0, \infty\}$$. One observes that $$\mathrm{Aut}_{\mathcal{E}}$$ is equivariant under the action of $$V := I(1)/I(2)$$ with respect to a generic character $$\Psi: V \to \mathbb{G}_a$$. One the other hand, it is also the clean extension of its restriction $$\mathcal{L}_\Psi$$ to an open substack $$\overset{\circ}{\mathrm{Bun}_{\mathcal{G}}} \simeq V$$; here $$\mathcal{L}_\Psi$$ denotes the pull-back of the exponential $$D$$-module on $$\mathbb{G}_a$$ through $$\Psi$$. One then deduces that the Hecke eigenvalue of $$\mathrm{Aut}_{\mathcal{E}}$$ coincides with the Frenkel-Gross connection, as desired.
In order to implement these ideas, the author enhances Beilinson-Drinfeld’s quantization of the Hitchin integrable system to the case of certain non-constant group schemes. He also studies the images of the duals of Moy-Prasad subgroups under the local Hitchin map. These results are technical and might be useful for other purposes in geometric Langlands program.

##### MSC:
 14D24 Geometric Langlands program (algebro-geometric aspects) 22E57 Geometric Langlands program: representation-theoretic aspects
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##### References:
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