Bessel $F$-isocrystals for reductive 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 ${\mathrm{Be}}_{\check{G}}^{†}$ 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-Ngô-Yun. By exploring the structure of the underlying differential equation, we calculate the monodromy group of ${\mathrm{Be}}_{\check{G}}^{†}$ when $\check{G}$ is almost simple (which recovers the calculation of monodromy group of ${\mathrm{Kl}}_{\check{G}}$ due to Katz and Heinloth-Ngô-Yun), and establish functoriality between different Kloosterman $\check{G}$-local systems as conjectured by Heinloth-Ngô-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 $G_2$.

arXiv comment71 pages. A few typos corrected. Add Remark 4.5.8 on local monodromy at infinity

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Cited by in corpus (3)

• Airy sheaves for reductive groups
• Local parameters of supercuspidal representations
• Newton polygons for $L$-functions of generalized Kloosterman sums