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# Pure and Applied Mathematics Quarterly

## Volume 16 (2020)

### Number 5

### Symplectic coordinates on $\operatorname{PSL}_3(\mathbb{R})$-Hitchin components

Pages: 1321 – 1386

DOI: https://dx.doi.org/10.4310/PAMQ.2020.v16.n5.a1

#### Authors

#### Abstract

Goldman parametrizes the $\operatorname{PSL}_3(\mathbb{R})$-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 $\operatorname{PSL}_3(\mathbb{R})$-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.

#### Keywords

Hitchin component, Goldman coordinates, Darboux coordinates

#### 2010 Mathematics Subject Classification

Primary 57M50. Secondary 53D30.

S. Choi and H. Jung were supported in part by NRF-2016R1D1A1B03932524.

Received 31 January 2020

Accepted 15 March 2020

Published 17 February 2021