### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 055, 11 pages      arXiv:1507.08365      https://doi.org/10.3842/SIGMA.2016.055

### A Family of Finite-Dimensional Representations of Generalized Double Affine Hecke Algebras of Higher Rank

Yuchen Fu and Seth Shelley-Abrahamson
Department of Mathematics, Massachusetts Institute of Technology, 182 Memorial Drive, Cambridge, MA 02139, USA

Received April 20, 2016, in final form June 11, 2016; Published online June 14, 2016

Abstract
We give explicit constructions of some finite-dimensional representations of generalized double affine Hecke algebras (GDAHA) of higher rank using $R$-matrices for $U_q(\mathfrak{sl}_N)$. Our construction is motivated by an analogous construction of Silvia Montarani in the rational case. Using the Drinfeld-Kohno theorem for Knizhnik-Zamolodchikov differential equations, we prove that the explicit representations we produce correspond to Montarani's representations under a monodromy functor introduced by Etingof, Gan, and Oblomkov.

Key words: generalized double affine Hecke algebra; $R$-matrix; Drinfeld-Kohno theorem.

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