Rigid connections on $\mathbb{P}^1$ via the Bruhat-Tits building

TL;DR

This paper classifies cohomologically rigid $G$-connections on the projective line with specific irregular and regular singularities, extending the work of Frenkel and Gross through the application of Bruhat-Tits building theory.

Contribution

It generalizes the classification of rigid connections by incorporating irregular homogeneous toral singularities associated with Coxeter tori using the framework of fundamental strata.

Findings

Characterization of all rigid connections with specified singularities.

Extension of Frenkel-Gross work to broader class of connections.

Application of Bruhat-Tits building theory to connection classification.

Abstract

We apply the theory of fundamental strata of Bremer and Sage to find cohomologically rigid $G$-connections on the projective line, generalising the work of Frenkel and Gross. In this theory, one studies the leading term of a formal connection with respect to the Moy-Prasad filtration associated to a point in the Bruhat-Tits building. If the leading term is regular semisimple with centraliser a (not necessarily split) maximal torus $S$, then we have an $S$-toral connection. In this language, the irregular singularity of the Frenkel-Gross connection gives rise to the homogenous toral connection of minimal slope associated to the Coxeter torus $\mathcal{C}$. In the present paper, we consider connections on $\mathbb{G}_m$ which have an irregular homogeneous $\mathcal{C}$-toral singularity at zero of slope $i/h$, where $h$ is the Coxeter number and $i$ is a positive integer coprime to $h$, and a regular singularity at infinity with unipotent monodromy. Our main result is the characterisation of all such connections which are rigid.