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In 2007, Alekseev-Meinrenken proved that there exists a Ginzburg-Weinsteindiffeomorphism from the dual Lie algebra ${\rm u}(n)^*$ to the dual Poisson Liegroup $U(n)^*$ compatible with the Gelfand-Zeitlin integrable systems. In thispaper, we explicitly construct such diffeomorphisms via Stokes phenomenon andBoalch's dual exponential maps. Then we introduce a relative version of theGinzburg-Weinstein linearization motivated by irregular Riemann-Hilbertcorrespondence, and generalize the results of Enriquez-Etingof-Marshall to thisrelative setting. In particular, we prove the connection matrix for a certainirregular Riemann-Hilbert problem satisfies a relative gauge transformationequation of the Alekseev-Meinrenken dynamical r-matrices. This gauge equationis then derived as the semiclassical limit of the relative Drinfeld twistequation.

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