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 D₄-type GDAHA to representations of the E₆-type one for specialised parameters. Then, under no restrictions on the parameters, we construct embeddings of both GDAHAs of type D₄ and E₆ into matrix algebras over quantum cluster X-varieties, thus linking to the theory of higher Teichmüller spaces. For E₆, the two explicit representations we provide over distinct quantum tori are shown to be related by quiver reductions and mutations.