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

Masoud Kamgarpour, Bailey Whitbread

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.

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

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.
Paper Structure (19 sections, 10 theorems, 18 equations, 4 tables)

This paper contains 19 sections, 10 theorems, 18 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Structure of the text
  3. Acknowledgements
  4. Toral connections
  5. Basic definitions
  6. Riemann--Hilbert Correspondence
  7. The Deligne--Simpson problem
  8. Nice braids
  9. Isoclinic connections
  10. Springer elements
  11. The braid associated to an isoclinic class
  12. Proof of Theorem \ref{['t:main']}
  13. Rigid isoclinic connections
  14. Explicit realisation
  15. Minimal unipotent classes for isoclinic connections
  16. ...and 4 more sections

Key Result

Theorem 1

There exists a (necessarily unique) unipotent conjugacy class $\mathcal{C}_\nu\subset G$ such that there is an isoclinic $G$-connection of slope $\nu$ and unipotent monodromy $\mathcal{C}$ if and only if $\mathcal{C} \ge \mathcal{C}_\nu$.Here $\geq$ denotes the usual partial order on unipotent conju

Theorems & Definitions (17)

  • Theorem 1
  • Theorem 2
  • Proposition 3
  • proof
  • Lemma 4
  • proof
  • Theorem 5: Lusztig
  • Theorem 6
  • proof
  • Proposition 7
  • ...and 7 more