Length functions of Hitchin representations

Given a Hitchin representation $\rho \text{\hspace{0.17em}⁣}:{\pi }_1(S)\to {\text{PSL}}_n(\mathbb{R})$, we construct $n$ continuous functions $\ell_i^\rho \colon \mathcal \CH(S) \to $\mathbb{R}$$definedonthespaceofH\ddot{o}ldergeodesiccurrents$\CH(S)$ such that, for a closed, oriented curve $\gamma$ in $S$, the $i$–th eigenvalue of the matrix $\rho (\gamma )\in {\text{PSL}}_n(\mathbb{R})$ is of the form $\pm $\mathrm{e}\mathrm{x}\mathrm{p}$, \ell_i^\rho(\gamma)$: such functions generalize to higher rank Thurston’s length function of Fuchsian re\presentations. Identities, diffe\rentiability properties of these lengths ${\mathrm{\ell }}_i^{\rho }$, as well as applications to eigenvalue estimates, are also considered.