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arXiv · 2307.06803

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

Davide Dal Martello·Marta Mazzocco
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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.

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Davide Dal Martello, Marta Mazzocco. 2023-07-13. Generalized double affine Hecke algebras, their representations, and higher Teichm\"uller theory. https://doi.org/10.1016/j.aim.2024.109763

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