Counting points on braid varieties and the Deligne--Simpson problem

TL;DR

This paper solves the isoclinic Deligne--Simpson problem for exceptional groups by counting points on braid varieties over finite fields, leading to new rigid irregular connections and advancing the understanding of braid varieties.

Contribution

It completes the isoclinic Deligne--Simpson problem for exceptional groups and introduces a novel counting method using braid varieties and finite fields.

Findings

Solved the isoclinic Deligne--Simpson problem for exceptional groups.

Established a new approach using point counting on braid varieties.

Produced new examples of rigid irregular connections.

Abstract

We solve the isoclinic Deligne--Simpson problem for exceptional groups, completing a program initiated by Sage et al. and Jakob--Yun. As a by-product, we obtain new examples of physically rigid irregular connections on the projective line. Our approach uses the Riemann--Hilbert correspondence to reduce the problem to determining the non-emptiness of certain braid varieties associated to periodic braids. We show that this can be achieved by counting points over finite fields. Our approach is inspired by Lusztig's construction of a map from conjugacy classes in the Weyl group to unipotent classes.