$p$-curvature of periodic pencils of flat connections

Abstract

In arXiv:2401.00636 we introduced the notion of a periodic pencil of flat connections on a smooth variety $X$. Namely, a pencil is a linear family of flat connections $\nabla(s_1,...,s_n)=d-\sum_{i=1}^r\sum_{j=1}^ns_jB_{ij}dx_i,$ where $\lbrace x_i\rbrace$ are coordinates on $X$ and $B_{ij}: X\to {\rm Mat}_N$ are matrix-valued regular functions. A pencil is periodic if it is generically invariant under the shifts $s_j\mapsto s_j+1$ up to isomorphism. In this paper we show that in characteristic $p>0$, the $p$-curvature operators $\lbrace C_i,1\le i\le r\rbrace$ of a periodic pencil $\nabla$ are isospectral to the commuting endomorphisms $C_i^*:=\sum_{j=1}^n (s_j-s_j^p)B_{ij}^{(1)}$, where $B_{ij}^{(1)}$ is the Frobenius twist of $B_{ij}$. Using the results of arXiv:2401.00636, this allows us to compute the eigenvalues of the $p$-curvature for many important examples of pencils of flat connections, including Knizhnik-Zamolodchikov (KZ), Casimir, and Dunkl connections, their confluent limits, and equivariant quantum connections for conical symplectic resolutions with finitely many torus fixed points. In particular, for rational values of parameters these eigenvalues are zero, so the connections are globally nilpotent. We also show that every periodic pencil has regular singularites and its residues have rational eigenvalues for rational values of parameters. In particular, this holds for the aforementioned quantum connections if they have rational coefficients. Also we generalize these results to irregular pencils (KZ, Casimir, Dunkl, and Toda), and relate them in the Dunkl case to representations of rational Cherednik algebras. Finally, we extend our main result to pseudo-pencils and discuss the generalization to difference equations.