Twisted Euler transform of differential equations with an irregular singular point

TL;DR

This paper generalizes the Euler transform to the twisted Euler transform for differential equations with irregular singular points, linking these transformations to Weyl group actions of Kac-Moody Lie algebras.

Contribution

It introduces the twisted Euler transform and explicitly describes its effects on irregular singular differential equations, connecting these to Lie algebra symmetries.

Findings

Explicit description of local data changes under twisted Euler transforms

Connection between twisted Euler transforms and Weyl group actions

Application to differential equations with irregular rank 2 singularity

Abstract

N. Katz introduced the notion of the middle convolution on local systems. This can be seen as a generalization of the Euler transform of Fuchsian differential equations. In this paper, we consider the generalization of the Euler transform, the twisted Euler transform, and apply this to differential equations with irregular singular points. In particular, for differential equations with an irregular singular point of irregular rank 2 at $x=\infty$, we describe explicitly changes of local datum caused by twisted Euler transforms. Also we attach these differential equations to Kac-Moody Lie algebras and show that twisted Euler transforms correspond to the action of Weyl groups of these Lie algebras.