D. Dal Martello and M. Mazzocco
Generalised double affine Hecke algebras, their representations, and higher Teichmüller theory
Advances in Mathematics (2024) 10.1016/j.aim.2024.109763

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.