Nonstandard representations of type C affine Hecke algebra from K-operators

Abstract

We construct nonstandard finite-dimensional representations of type C affine Hecke algebra from the viewpoint of quantum integrable models. There exists two classes of nonstandard solutions to the Yang-Baxter equation called the Cremmer-Gervais and Jordanian R-matrices. These R-matrices also satisfy the Hecke-relation, thus can be used to construct nonstandard finite-dimensional representations of type A affine Hecke algebra. We construct the corresponding nonstandard representations for type C affine Hecke algebra by explicitly constructing solutions to the reflection equation under the Hecke relation. We achieve this by taking the finite-dimensional representations and deBaxterizing the K-operators acting on the infinite-dimensional function space, taking advantage of the fact that the Cremmer-Gervais and Jordanian R-matrices can be obtained from the R-operator.