*preprint*

**Inserted:** 2 dec 2023

**Last Updated:** 2 dec 2023

**Year:** 2023

**Abstract:**

The aim of this paper is to show the existence and give an explicit description of a pseudo-Riemannian metric and a symplectic form on the $\mathbb{P}\mathrm{S}\mathrm{L}(3,\mathbb{R})$-Hitchin component, both compatible with Labourie and Loftin's complex structure. In particular, they give rise to a mapping class group invariant pseudo-K\"ahler structure on a neighborhood of the Fuchsian locus, which restricts to a multiple of the Weil-Petersson metric on Teichm\"uller space. Finally, generalizing a previous result in the case of the torus, we prove the existence of an Hamiltonian $S^1$-action.