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 Ẽ6 -type one for specialised parameters.Then, under no restrictions on the parameters, we construct embeddings of both GDAHAs of type D4 and Ẽ6 into matrix algebras over quantum cluster X -varieties, thus linking to the theory of higher Teichmüller spaces.For Ẽ6 , the two explicit representations we provide over distinct quantum tori are shown to be related by quiver reductions and mutations.