Conformal blocks for Galois covers of algebraic curves

Abstract

We study the spaces of twisted conformal blocks attached to a $Γ$-curve $Σ$ with marked $Γ$-orbits and an action of $Γ$ on a simple Lie algebra $\mathfrak{g}$, where $Γ$ is a finite group. We prove that if $Γ$ 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 $Γ$-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 $\mathscr{G}$ be the parahoric Bruhat-Tits group scheme on the quotient curve $Σ/Γ$ obtained via the $Γ$-invariance of Weil restriction associated to $Σ$ 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 $\mathscr{G}$-torsors on $Σ/Γ$ when the level $c$ is divisible by $|Γ|$ (establishing a conjecture due to Pappas-Rapoport).

Theorems & Definitions (123)

• Lemma 2.1
• proof
• Corollary 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Definition 3.1