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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 $G_2$.
We prove that for an indecomposable convergent or overconvergent F-isocrystal on a smooth irreducible variety over a perfect field of characteristic p, the gap between consecutive slopes at the generic point cannot exceed 1. (This may be thought of a
Let LG be an algebraic loop group associated to a reductive group G. A fundamental stratum is a triple consisting of a point x in the Bruhat-Tits building of LG, a nonnegative real number r, and a character of the corresponding depth r Moy-Prasad sub
We study local-global principles for torsors under reductive linear algebraic groups over semi-global fields; i.e., over one variable function fields over complete discretely valued fields. We provide conditions on the group and the semiglobal field
Let G be a split reductive group. We introduce the moduli problem of bundle chains parametrizing framed principal G-bundles on chains of lines. Any fan supported in a Weyl chamber determines a stability condition on bundle chains. Its moduli stack pr
This paper generalizes the classical theory of Newton polygons from the case of general linear groups to the case of split reductive groups. It also gives a root-theoretic formula for dimensions of Newton strata in the adjoint quotients of reductive groups.