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. As a corollary, we establish the de Rham analogue of a conjecture of Heinloth, Nĝo, and Yun for G = SLn.
Publication Source (Journal or Book title)
American Journal of Mathematics
Kamgarpour, M., & Sage, D. (2019). A geometric analogue of a conjecture of gross and reeder. American Journal of Mathematics, 141 (5), 1457-1476. https://doi.org/10.1353/ajm.2019.0038