A Deligne-Simpson problem for irregular $G$-connections over $\mathbb{P}^{1}$

TL;DR

This paper establishes algebraic and geometric criteria for the existence of irregular $G$-connections on $P^1$ with specified local data, solving the Deligne-Simpson problem for classical groups and certain slopes.

Contribution

It introduces new algebraic and geometric conditions for $G$-connections with irregular types, providing complete solutions for classical groups and specific slopes, including classification of rigid connections.

Findings

Criteria in terms of rational Cherednik algebra modules and affine Springer fibers.

Complete solutions to the isoclinic Deligne-Simpson problem for classical groups.

Classification of cohomologically rigid connections, including new cases in types B, C, and F4.

Abstract

We give an algebraic and a geometric criterion for the existence of $G$-connections on $\mathbb{P}^{1}$ with prescribed irregular type with equal slope at $\infty$ (isoclinic) and with regular singularity of prescribed residue at $0$. The algebraic criterion is in terms of an irreducible module of the rational Cherednik algebra, and the geometric criterion is in terms of affine Springer fibers. We use these criteria to give complete solutions to the isoclinic Deligne-Simpson problem for classical groups, and for arbitrary $G$ when the slope at $\infty$ has Coxeter number as the denominator. Among our solutions, we classify the cohomologically rigid connections, and obtain new cases in types $B,C$ and $F_4$.