Generalized double affine Hecke algebras, their representations, and higher Teichmüller theory

Generalized double affine Hecke algebras (GDAHA) are flat deformations of the group algebras of 2-dimensional crystallografic groups associated to star-shaped simply laced affine Dynkin diagrams. In this paper, we construct a functor sending representations of the $\tilde D_4$-type GDAHA to representations of a specialization of the $\tilde E_6$-type one. Moreover, we construct embeddings of the GDAHAs of type $\tilde D_4$ and $\tilde E_6$ into matrix algebras over the coordinate ring of quantum cluster $\mathcal{X}$-varieties, thus linking to the theory of higher Teichmüller spaces.