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

TL;DR

This paper constructs explicit finite-dimensional representations of generalized double affine Hecke algebras of higher rank using R-matrices, linking them to known rational case representations via monodromy functors.

Contribution

It provides a new explicit construction of GDAHA representations of higher rank, extending previous rational case work and connecting algebraic and monodromy perspectives.

Findings

Explicit finite-dimensional GDAHA representations constructed

Connections established between algebraic and monodromy representations

Extension of rational case constructions to higher rank

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.