Symplectic coordinates on PSL3(R)-Hitchin components

Abstract

Goldman parametrizes the PSL 3 (R) -Hitchin component of a closed oriented hyperbolic surface of genus g by 16g−16 parameters. Among them, 10g−10 coordinates are canonical. We prove that the PSL 3 (R)-Hitchin component equipped with the Atiyah–Bott–Goldman symplectic form admits a global Darboux coordinate system such that the half of its coordinates are canonical Goldman coordinates. To this end, we show a version of the action-angle principle and the Zocca-type decomposition formula for the symplectic form of H. Kim and Guruprasad–Huebschmann–Jeffrey-Weinstein given to symplectic leaves of the Hitchin component.