Conformal blocks for Galois covers of algebraic curves

Abstract

We study the spaces of twisted conformal blocks attached to a $\hat{I}“$-curve $\hat{I}\pounds$ with marked $\hat{I}“$-orbits and an action of $\hat{I}“$ on a simple Lie algebra $\mathfrak{g}$, where $\hat{I}“$ is a finite group. We prove that if $\hat{I}“$ stabilizes a Borel subalgebra of $\mathfrak{g}$, then Propagation Theorem and Factorization Theorem hold. We endow a flat projective connection on the sheaf of twisted conformal blocks attached to a smooth family of pointed $\hat{I}“$-curves; in particular, it is locally free. We also prove that the sheaf of twisted conformal blocks on the stable compactification of Hurwitz stack is locally free. Let $\mathcal{G}$ be the parahoric Bruhat-Tits group scheme on the quotient curve $\hat{I}\pounds /\hat{I}“$ obtained via the $\hat{I}“$-invariance of Weil restriction associated to $\hat{I}\pounds$ and the simply-connected simple algebraic group $G$ with Lie algebra $\mathfrak{g}$. We prove that the space of twisted conformal blocks can be identified with the space of generalized theta functions on the moduli stack of quasi-parabolic $\mathcal{G}$-torsors on $\hat{I}\pounds /\hat{I}“$ when the level $c$ is divisible by $|\hat{I}“|$ (establishing a conjecture due to Pappas-Rapoport).

This new version of the paper fixes an error in the statement of Lemma 8.3 in the published version of the paper