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Stokes phenomenon of Kloosterman and Airy connections

Andreas Hohl, Konstantin Jakob

Abstract

We define categories of Stokes filtered and Stokes graded $G$-local systems for reductive groups $G$ and use the formalism of Tannakian categories to show that they are equivalent to the category of $G$-connections. We then use the interpretation of moduli spaces of Stokes filtered $G$-local systems as braid varieties to prove physical rigidity of two well-known families of cohomologically rigid connections, the Kloosterman and Airy connections. In the Kloosterman case, our proof relies on Steinberg's cross-section.

Stokes phenomenon of Kloosterman and Airy connections

Abstract

We define categories of Stokes filtered and Stokes graded -local systems for reductive groups and use the formalism of Tannakian categories to show that they are equivalent to the category of -connections. We then use the interpretation of moduli spaces of Stokes filtered -local systems as braid varieties to prove physical rigidity of two well-known families of cohomologically rigid connections, the Kloosterman and Airy connections. In the Kloosterman case, our proof relies on Steinberg's cross-section.
Paper Structure (27 sections, 24 theorems, 73 equations, 1 table)

This paper contains 27 sections, 24 theorems, 73 equations, 1 table.

Table of Contents

  1. Introduction
  2. Kloosterman connections
  3. Generalized Kloosterman connections
  4. Airy connections
  5. Organization of the paper
  6. Setting and notation
  7. $G$-local systems
  8. The group $G$ and related data
  9. The base curve
  10. Stokes filtrations and Stokes gradings
  11. Exponential local system and irregular classes
  12. Stokes directions, singular directions, and the exponential dominance order
  13. Positive and negative weights of $\Theta$
  14. Stokes conditions
  15. Stokes filtrations and Stokes gradings
  16. ...and 12 more sections

Key Result

Theorem 1.1.1

Assume $G$ is a simple complex algebraic group and let $\mathcal{C}$ be a regular conjugacy class in $G$. Moreover, assume $G$ is simply connected or $\mathcal{C}$ is the regular unipotent class. Then, the Kloosterman $G$-connection with local monodromy at $0$ in $\mathcal{C}$ is physically rigid.

Theorems & Definitions (61)

  • Theorem 1.1.1: see Theorem \ref{['thm:KloostermanPhysicalRigidity']}
  • Theorem 1.2.1: see Corollary \ref{['cor:genkloost']}
  • Theorem 1.3.1: see Theorem \ref{['thm:AiryPhysicalRigidity']}
  • Definition 2.1.1
  • Definition 3.1.1: BY
  • Lemma 3.1.2
  • proof
  • Definition 3.1.3
  • Remark 3.1.4
  • Definition 3.2.1
  • ...and 51 more