Symplectic coordinates on PSL3(R)-Hitchin components

Abstract

Goldman parametrizes the PSL3(R)-Hitchin component of a closed oriented hyperbolic surface of genus g by 16(g) - 16 parameters. Among them, 10(g) - 10 coordinates are canonical. We prove that the PSL3(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.