# Stokes phenomenon, Gelfand-Zeitlin systems and relative Ginzburg-Weinstein linearization

## TL;DR

This paper constructs explicit Ginzburg-Weinstein diffeomorphisms using Stokes phenomena, introduces a relative version linked to irregular Riemann-Hilbert problems, and generalizes existing results to this new setting.

## Contribution

It provides explicit constructions of Ginzburg-Weinstein diffeomorphisms via Stokes phenomena and extends the theory to a relative setting inspired by irregular Riemann-Hilbert correspondence.

## Key findings

- Explicit Ginzburg-Weinstein diffeomorphisms constructed
- Connection matrix satisfies a relative gauge transformation equation
- Semiclassical limit relates to the relative Drinfeld twist equation

## Abstract

In 2007, Alekseev-Meinrenken proved that there exists a Ginzburg-Weinstein diffeomorphism from the dual Lie algebra ${\rm u}(n)^*$ to the dual Poisson Lie group $U(n)^*$ compatible with the Gelfand-Zeitlin integrable systems. In this paper, we explicitly construct such diffeomorphisms via Stokes phenomenon and Boalch's dual exponential maps. Then we introduce a relative version of the Ginzburg-Weinstein linearization motivated by irregular Riemann-Hilbert correspondence, and generalize the results of Enriquez-Etingof-Marshall to this relative setting. In particular, we prove the connection matrix for a certain irregular Riemann-Hilbert problem satisfies a relative gauge transformation equation of the Alekseev-Meinrenken dynamical r-matrices. This gauge equation is then derived as the semiclassical limit of the relative Drinfeld twist equation.