Non-Stationary Difference Equation and Affine Laumon Space II: Quantum Knizhnik-Zamolodchikov Equation

We show that Shakirov's non-stationary difference equation, when it is truncated, implies the quantum Knizhnik-Zamolodchikov (q-KZ) equation for U-v(A(1)((1))) with 1 generic spins. Namely, we can tune mass parameters so that the Hamiltonian acts on the space of finite Laurent polynomials. Then the representation matrix of the Hamiltonian agrees with the R-matrix, or the quantum 6j symbols. On the other hand, we prove that the K theoretic Nekrasov partition function from the affine Laumon space is identified with the well-studied Jackson integral solution to the q-KZ equation. Combining these results, we establish that the affine Laumon partition function gives a solution to Shakirov's equation, which was a conjecture in our previous paper. We also work out the base-fiber duality and four-dimensional limit in relation with the q-KZ equation.