Solutions of the sl 2 ${\mathfrak {sl}_2}$ qKZ equations modulo an integer

Abstract

We study the qKZ difference equations with values in the $n$th tensor power of the vector ${\mathrm{sl}}_2$ representation $V$, variables $z_1,\cdots ,z_n$, and integer step $\kappa$. For any integer $N$ relatively prime to the step $\kappa$, we construct a family of polynomials $f_r\left(z\right)$ in variables $z_1,\cdots ,z_n$ with values in $V^{\otimes n}$ such that the coordinates of these polynomials with respect to the standard basis of $V^{\otimes n}$ are polynomials with integer coefficients. We show that $f_r\left(z\right)$ satisfy the qKZ equations modulo $N$. Polynomials $f_r\left(z\right)$ are modulo $N$ analogs of the hypergeometric solutions of the qKZ given in the form of multidimensional Barnes integrals.