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

Abstract

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.

51 pages, 23 figures; final accepted version

References in corpus (2)

• The quantum dilogarithm and representations quantum cluster varieties
• The Relationship between Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case

Cited by in corpus (1)

• Okamoto's symmetry on the representation space of the sixth Painlevé equation