# $p$-curvature operators and Satake-type phenomenon for $\frak{sl}_2$ KZ equations with $\kappa=\pm 2$

## TL;DR

This paper investigates the $rak{sl}_2$ KZ equations at specific parameters, establishing a Satake-type correspondence to analyze $p$-curvature operators and solution spaces in characteristic $p$, revealing new phenomena about solution generation.

## Contribution

It introduces a Satake-type correspondence for $rak{sl}_2$ KZ equations at $
abla= ext{pm} 2$, enabling analysis of $p$-curvature and solution spaces in characteristic $p$ with novel examples.

## Key findings

- Not all solutions are generated by $p$-hypergeometric solutions in characteristic $p$ for $
abla=2$.
- Established a Satake-type correspondence for $rak{sl}_2$ KZ equations.
- Analyzed the dimension of solution spaces in characteristic $p$.

## Abstract

The $\frak{sl}_2$ KZ differential equations with values in the tensor power of the fundamental representation with parameter $\kappa=\pm 2$ are considered. A Satake-type correspondence is established over complex numbers and subsequently reduced to finite characteristic. This correspondence enables the study of the KZ equations on the lower weight subspaces of the tensor power in terms of the wedge powers of the weight subspace of the weight just below the highest weight. We apply this approach to analyze the $p$-curvature operators associated with our KZ equations, evaluate the dimension of the solution space in characteristic $p$, and determine whether all solutions are generated by the so-called $p$-hypergeometric solutions. In particular, we show that not all solutions of the KZ equations with $\kappa=2$ in characteristic $p$ are generated by $p$-hypergeometric solutions. Previously, no such examples were known.