Stokes phenomenon, Gelfand–Zeitlin systems and relative Ginzburg–Weinstein linearization

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In 2007, Alekseev–Meinrenken proved that there exists a Ginzburg–Weinstein diffeomorphism from the dual Lie algebra 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.