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 crystallographic groups associated to star-shaped simply laced affine Dynkin diagrams. In this paper, we first construct a functor that sends representations of the ${\tilde{D}}_4$-type GDAHA to representations of the ${\tilde{E}}_6$-type one for specialised parameters. Then, under no restrictions on the parameters, we construct embeddings of both GDAHAs of type ${\tilde{D}}_4$ and ${\tilde{E}}_6$ into matrix algebras over quantum cluster $\mathcal{X}$-varieties, thus linking to the theory of higher Teichmüller spaces. For ${\tilde{E}}_6$, the two explicit representations we provide over distinct quantum tori are shown to be related by quiver reductions and mutations.