Level structure on a G-bundle

Definition

Let $E$ be a principal $G$ -bundle on a smooth curve $X$ , let $x\in X$ , and choose a subgroup $K\subseteq G(\mathcal O_x)$ of the positive loop group. The trivializations of $E$ on the formal disc at $x$ form a $G(\mathcal O_x)$ -torsor. A $K$ -level structure on $E$ at $x$ is a reduction of this torsor to $K$ , equivalently a section of its quotient by $K$ .

Standard choices

$K=\{1\}$ gives a full formal trivialization.
If $K$ is the inverse image of a Borel subgroup under $G(\mathcal O_x)\to G$ , the datum is Iwahori level.
Bruhat–Tits parahoric subgroups give parahoric level.
Congruence subgroups give successively deeper level structures.

Automorphic moduli

Replacing $\operatorname{Bun}_G(X)$ by the stack of bundles with a fixed $K$ -level structure changes the automorphic sheaf category. Which spectral ramification condition corresponds to it depends on $K$ , the equivariance imposed on automorphic sheaves, and any chosen character of a deeper filtration quotient.

References

Georgios Pappas and Michael Rapoport, “Twisted loop groups and their affine flag varieties,” Advances in Mathematics 219 (2008), 118–198. arXiv.