A geometric analogue of a conjecture of Gross and Reeder

TL;DR

This paper proves that for irregular flat G-bundles on the formal punctured disk, the irregularity of the adjoint connection is at least the rank of G, and characterizes those with minimal irregularity.

Contribution

It establishes a geometric analogue of Gross and Reeder's conjecture and identifies Frenkel-Gross connections as those with minimal irregularity.

Findings

Irregularity of adjoint connection ≥ rank of G

Frenkel-Gross connections have minimal irregularity

Provides a geometric perspective on Gross-Reeder conjecture

Abstract

Let G be a simple complex algebraic group. We prove that the irregularity of the adjoint connection of an irregular flat G-bundle on the formal punctured disk is always greater than or equal to the rank of G. This can be considered as a geometric analogue of a conjecture of Gross and Reeder. We will also show that the irregular connections with minimum adjoint irregularity are precisely the (formal) Frenkel-Gross connections.