Cluster realizations of Weyl groups and higher Teichm\"uller theory

For a symmetrizable Kac-Moody Lie algebra $\mathfrak{g}$, we construct a family of weighted quivers $Q_m(\mathfrak{g})$ ($m \geq 2$) whose cluster modular group $\Gamma_{Q_m(\mathfrak{g})}$ contains the Weyl group $W(\mathfrak{g})$ as a subgroup. We compute explicit formulae for the corresponding cluster $\mathcal{A}$- and $\mathcal{X}$-transformations. As a result, we obtain green sequences and the cluster Donaldson-Thomas transformation for $Q_m(\mathfrak{g})$ in a systematic way when $\mathfrak{g}$ is of finite type. Moreover if $\mathfrak{g}$ is of classical finite type with the Coxeter number $h$, the quiver $Q_{kh}(\mathfrak{g})$ ($k \geq 1$) is mutation-equivalent to a quiver encoding the cluster structure of the higher Teichm\"uller space of a once-punctured disk with $2k$ marked points on the boundary, up to frozen vertices. This correspondence induces the action of direct products of Weyl groups on the higher Teichm\"uller space of a general marked surface. We finally prove that this action coincides with the one constructed in [GS18] from the geometrical viewpoint.