Quantum symmetric pairs and representations of double affine Hecke algebras of type $C^\vee C_n$

TL;DR

This paper constructs representations of affine and double affine braid groups and Hecke algebras of type C^vee C_n using quantum symmetric pairs, generalizing previous results and providing new quantizations.

Contribution

It introduces a new method to build representations of these algebraic structures based on quantum symmetric pairs, extending prior work to type BC.

Findings

Provides a quantization of existing representations by Etingof, Freund, and Ma

Generalizes results to type BC for the first time

Constructs explicit representations for affine and double affine braid groups

Abstract

We build representations of the affine and double affine braid groups and Hecke algebras of type $C^\vee C_n$, based upon the theory of quantum symmetric pairs $(U,B)$. In the case $U=U_q(gl_N)$, our constructions provide a quantization of the representations constructed by Etingof, Freund and Ma in arXiv:0801.1530, and also a type $BC$ generalization of the results in arXiv:0805.2766.