A Deligne-Simpson problem for irregular G-connections over ℙ¹

2023/01/26 by Konstantin Jakob, Jakob, Konstantin, Zhiwei Yun +1
#math.AG #math.RT

paper · pdf · doi:10.48550/arxiv.2301.10967

Abstract

We give an algebraic and a geometric criterion for the existence of G-connections on ℙ¹ with prescribed irregular type with equal slope at ∞ (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 ∞ has Coxeter number as the denominator. Among our solutions, we classify the cohomologically rigid connections, and obtain new cases in types B,D and F₄.

Related

• Explicit constructions of connections on the projective line with a maximally ramified irregular singularity
• Deformation of moduli spaces of meromorphic G-connections on ℙ¹ via unfolding of irregular singularities
• A rigid irregular connection on the projective line
• Meromorphic connections on the projective line with specified local behavior
• Rigid G-connections and nilpotency of p-curvatures