Generalized double affine Hecke algebras, their representations, and higher Teichm\"uller theory

TL;DR

This paper explores generalized double affine Hecke algebras (GDAHA), establishing connections between their representations and higher Teichmüller theory through embeddings into quantum cluster varieties and explicit representation constructions.

Contribution

It constructs a functor linking GDAHA representations of different types and embeds these algebras into quantum cluster varieties, connecting algebraic and geometric frameworks.

Findings

Constructed a functor between $ ilde D_4$ and $ ilde E_6$ GDAHA representations.

Embedded GDAHA of types $ ilde D_4$ and $ ilde E_6$ into matrix algebras over quantum cluster $ ext{X}$-varieties.

Provided explicit representations of $ ilde E_6$ GDAHA related by quiver mutations.

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 Teichm\"uller 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.