标题： $p$-曲率算子和$\frak{sl}_2$ KZ方程的Satake型现象与$κ=\pm 2$ 显示英文标题

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

摘要： 考虑参数为$\kappa=\pm 2$的基本表示的张量幂中的$\frak{sl}_2$KZ 微分方程。 在复数上建立了类似 Satake 的对应关系，并随后降低到有限特征。 这种对应关系使得能够通过最高权下方的权子空间的楔积来研究张量幂的低权子空间上的 KZ 方程。 我们将这种方法应用于分析与我们的 KZ 方程相关的$p$曲率算子，计算特征$p$中解空间的维数，并确定所有解是否由所谓的$p$超几何解生成。 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. 显示英文摘要

摘要： 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.