Bessel F-isocrystals for reductive groups

Abstract

We construct the Frobenius structure on a rigid connection Be_Ğ on G_m for a split reductive group Ğ introduced by Frenkel–Gross. These data form a Ğ-valued overconvergent F-isocrystal Be^†_Ğ on G_(m,F_p), which is the p-adic companion of the Kloosterman Ğ-local system Kl_Ğ constructed by Heinloth–Ngô–Yun. By studying the structure of the underlying differential equation, we calculate the monodromy group of Be^†_Ğ when Ğ is almost simple (which recovers the calculation of monodromy group of Kl_Ğ due to Katz and Heinloth–Ngô–Yun), and prove a conjecture of Heinloth–Ngô–Yun on the functoriality between different Kloosterman Ğ-local systems. We show that the Frobenius Newton polygons of Kl_Ğ are generically ordinary for every Ğ and are everywhere ordinary on |G_(m,F_p)| when Ğ is classical or G₂.

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