An invariant subbundle of the KZ connection mod $p$ and reducibility of $\hat{sl}_2$ Verma modules mod $p$

Abstract: We consider the KZ differential equations over $\mathbb C$ in the case, when its multidimensional hypergeometric solutions are one-dimensional integrals. We also consider the same differential equations over a finite field $\mathbb F_p$. We study the space of polynomial solutions of these differential equations over $\mathbb F_p$, constructed in a previous work by V. Schechtman and the author. The module of these polynomial solutions defines an invariant subbundle of the associated KZ connection modulo $p$. We describe the algebraic equations for that subbundle and argue that the equations correspond to highest weight vectors of the associated $\hat{sl}_2$ Verma modules over the field $\mathbb F_p$.