Parahoric level structure

Definition

Let $G$ be a reductive group over the local field $k((t))$, and let $\mathcal G$ be the Bruhat–Tits parahoric group scheme on $\operatorname{Spec}k\lbrack\!\lbrack t\rbrack\!\rbrack$ attached to a facet of the Bruhat–Tits building. A parahoric level structure on a punctured-disc $G$-bundle is an extension to a $\mathcal G$-torsor on the formal disc.

Its formal-disc points $K=\mathcal G(k\lbrack\!\lbrack t\rbrack\!\rbrack)$ form the corresponding parahoric subgroup of $G(k((t)))$. When $K$ embeds in a chosen positive loop group, the same datum can be expressed as a reduction of the formal-frame torsor to $K$.