Frenkel-gross’ Irregular Connection and Heinloth-ngô-yun’s Are the Same

We show that the irregular connection on Gm constructed by Frenkel-Gross ([FG]) and the one constructed by Heinloth-Ngô-Yun ([HNY]) are the same, which confirms the Conjecture 2.14 of [HNY]. The proof is simple, modulo the big machinery of quantization of Hitchin’s integrable systems as developed by Beilinson-Drinfeld ([BD]). The idea is as follows. Let E be the irregular connection on Gm as constructed by Frenkel-Gross. It admits a natural oper form. We apply the machinery of Beilinson-Drinfeld to produce an automorphic D-module on the corresponding moduli space of G-bundles, with Hecke eigenvalue E . We show that this automorphic D-module is equivariant with respect to the unipotent group I(1)/I(2) (see [HNY] for the notation) against the non-degenerate additive character Ψ. By the uniqueness of such D-modules on the moduli space, one knows that the automorphic D-module constructed using the Beilinson-Drinfeld machinery is the same as the automorphic D-module explicitly constructed by Heinloth-Ngô-Yun. Since the irregular connection on Gm constructed in [HNY] is by definition the Hecke-eigenvalue of this automorphic D-module, it is the same as E .