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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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