A note on the bundle underlying Opers

TL;DR

This paper clarifies that for any simple Lie algebra, the bundle underlying a $rak{g}$-Oper on a smooth curve depends solely on the curve itself and is derived from the canonical automorphism bundle, simplifying existing understanding.

Contribution

It provides a clear proof that the underlying bundle of a $rak{g}$-Oper is determined only by the curve and is induced by the canonical automorphism bundle, enhancing transparency in the literature.

Findings

The underlying bundle depends only on the curve.

It is induced by the canonical automorphism bundle.

The proof clarifies a well-known fact.

Abstract

In this note we write down a proof of the following well known fact, in order to make the literature more transparent. Let $\mathfrak{g}$ be a simple Lie algebra, then for any smooth curve $C$, the bundle underlying any $\mathfrak{g}$-Oper depends only on the curve and it is induced by the canonical $\text{Aut}\, O$ bundle $\text{Aut}_C$ on $C$.