Quantization of Hitchin integrable system via positive characteristic

TL;DR

This paper demonstrates a simplified proof for the quantization of the Hitchin integrable system for G=GL(n) by leveraging positive characteristic methods and generic Langlands duality, connecting algebraic geometry and number theory.

Contribution

It provides a new, concise proof of Hitchin system quantization for G=GL(n) using positive characteristic reduction and Langlands duality, building on prior unpublished work.

Findings

Short proof of Hitchin system quantization for G=GL(n).

Reduction to positive characteristic simplifies complex geometric arguments.

Properties of p-curvature map on opers are established.

Abstract

In a celebrated unpublished manuscript Beilinson and Drinfeld quantize the Hitchin integrable system by showing that the global sections of critically twisted differential operators on the moduli stack of G-bundles on an algebraic curve is identified with the ring of regular functions on the space of G-opers; they deduce existence of an automorphic D-module corresponding to a local system carrying a structure of an oper. In this note we show for G=GL(n) that those results admit a short proof by reduction to positive characteristic, where they are deduced from generic Langlands duality established earlier by the first author and A. Braverman. The appendix contains a proof of some properties of the p-curvature map restricted to the space of opers.