Conformal blocks, parahoric torsors and Borel-Weil-Bott

TL;DR

This paper proves a conjecture relating to the cohomology of line bundles on moduli stacks of torsors over curves, establishing key properties like propagation of vacua and projective flatness, with implications for twisted vacua.

Contribution

It provides a proof of a conjecture for parahoric group schemes in characteristic zero and establishes foundational results on the cohomology of line bundles on moduli stacks.

Findings

Proof of Conjecture 3.7 in characteristic zero

Principle of propagation of vacua established

Cohomology of line bundles vanishes in all but one degree

Abstract

Let $X$ be a smooth projective curve over an algebraically closed field $k$. Let $\mathcal{G}$ be a parahoric group scheme on $X$ as in \cite{pr}. Via the principle of Hecke correspondences, we set-up relationships between the cohomology of lines bundles on various moduli stacks of torsors. This approach gives a proof of \cite[Conjecture 3.7]{pr} for group schemes $\mathcal G$ as above in characteristic zero. This further gives as a consequence, the principle of propagation of vacua. We give a direct proof of the independence of central charge on base points. Projective flatness is recovered as a corollary of Faltings construction of the Hitchin connection. Using C.Teleman's basic results (\cite{bwb}), we deduce the analogous result that cohomology of line bundles on the stack of principal $G$-bundles vanish in all degrees except possibly one. Results on twisted vacua \cite{hongkumar} are obtained as immediate consequences.