Connection Problems for Quantum Affine KZ Equations and Integrable Lattice Models

Cherednik attached to an affine Hecke algebra module a compatible system of difference equations, called quantum affine Knizhnik-Zamolodchikov (KZ) equations. In the case of a principal series module, we construct a basis of power series solutions of the quantum affine KZ equations. Relating the bases for different asymptotic sectors gives rise to a Weyl group cocycle, which we compute explicitly in terms of theta functions.

For the spin representation of the affine Hecke algebra of type C, the quantum affine KZ equations become the boundary qKZ equations associated to the Heisenberg spin-12 XXZ chain. We show that in this special case the results lead to an explicit 4-parameter family of elliptic solutions of the dynamical reflection equation associated to Baxter’s 8-vertex face dynamical R-matrix. We use these solutions to define an explicit 9-parameter elliptic family of boundary quantum Knizhnik-Zamolodchikov-Bernard (KZB) equations.