Bessel $F$-isocrystals for reductive groups

TL;DR

This paper constructs and analyzes Frobenius structures on certain rigid connections associated with split reductive groups, establishing their monodromy groups, functoriality, and ordinarity properties, extending previous work on Kloosterman local systems.

Contribution

It introduces a new construction of Frobenius structures on Be_{G} connections, computes their monodromy groups, and proves functoriality and ordinarity results for Kloosterman G-local systems.

Findings

Monodromy groups of Be_{G}^{ abla} are computed for almost simple G.

Functoriality between different Kloosterman G-local systems is established.

Frobenius Newton polygons are generically and everywhere ordinary for classical and G2 groups.

Abstract

We construct the Frobenius structure on a rigid connection $\mathrm{Be}_{\check{G}}$ on $\mathbb{G}_m$ for a split reductive group $\check{G}$ introduced by Frenkel-Gross. These data form a $\check{G}$-valued overconvergent $F$-isocrystal $\mathrm{Be}_{\check{G}}^{\dagger}$ on $\mathbb{G}_{m,\mathbb{F}_p}$, which is the $p$-adic companion of the Kloosterman $\check{G}$-local system $\mathrm{Kl}_{\check{G}}$ constructed by Heinloth-Ng\^o-Yun. By exploring the structure of the underlying differential equation, we calculate the monodromy group of $\mathrm{Be}_{\check{G}}^{\dagger}$ when $\check{G}$ is almost simple (which recovers the calculation of monodromy group of $\mathrm{Kl}_{\check{G}}$ due to Katz and Heinloth-Ng\^o-Yun), and establish functoriality between different Kloosterman $\check{G}$-local systems as conjectured by Heinloth-Ng\^o-Yun. We show that the Frobenius Newton polygons of $\mathrm{Kl}_{\check{G}}$ are generically ordinary for every $\check{G}$ and are everywhere ordinary on $|\mathbb{G}_{m,\mathbb{F}_p}|$ when $\check{G}$ is classical or $G_2$.