Moduli stack of G-bundles on a curve

Definition

Let $X$ be a smooth projective curve over $k$ , and let $G$ be a connected reductive algebraic group. The moduli stack of $G$ -bundles, $\operatorname{Bun}_G(X)$ , assigns to each $k$ -scheme $S$ the groupoid of principal $G_S$ -bundles on $X\times_k S$ .

Morphisms are bundle isomorphisms, so this construction retains automorphism groups rather than merely recording isomorphism classes. Under the stated hypotheses, $\operatorname{Bun}_G(X)$ is an algebraic stack locally of finite type over $k$ .

Automorphic role

The automorphic side of de Rham geometric Langlands is built from $D$ -modules on $\operatorname{Bun}_G(X)$ . Hecke correspondences modify a bundle at a point of $X$ and define functors on this category.

Examples and cautions

For $G=GL_n$ , principal $G$ -bundles are equivalent to rank- $n$ vector bundles. For $G=\mathbb G_m$ , $\operatorname{Bun}_G$ is the Picard stack. When $X$ is geometrically connected, the components for $GL_n$ are indexed by degree. More generally, over an algebraically closed field, the components for a connected reductive group are indexed by its algebraic fundamental group $\pi_1(G)$ ; degree is the resulting invariant for $GL_n$ and $\mathbb G_m$ .

The notation suppresses the fixed curve. It also suppresses derived enhancements used in the modern correspondence.

References

Vladimir Drinfeld and Carlos Simpson, “B-structures on G-bundles and local triviality,” Mathematical Research Letters 2 (1995), 823–829. DOI.
Jochen Heinloth, “Uniformization of $G$ -bundles,” Mathematische Annalen 347 (2010), 499–528. DOI.