Bessel $F$-isocrystals for reductive groups

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 $G_2$.