The affine Grassmannian as a presheaf quotient

Abstract: For a reductive group $G$ over a ring $A$, its affine Grassmannian $\mathrm{Gr}_G$ plays important roles in a wide range of subjects and is typically defined as the \'etale sheafification of the presheaf quotient $LG/L^+G$ of the loop group $LG$ by its positive loop subgroup $L^+G$. We show that the Zariski sheafification gives the same result. Moreover, for totally isotropic $G$ (for instance, for quasi-split $G$), we show that no sheafification is needed at all: $\mathrm{Gr}_G$ is already the presheaf quotient $LG/L^+G$, which seems new already in the classical case of $\mathrm{GL}_n$ over $\mathbb{C}$. For totally isotropic $G$, we also show that the affine Grassmannian may be formed using polynomial loops. We deduce all of these results from the study of $G$-torsors on $\mathbb{P}^1_A$ that is ultimately built on the geometry of $\mathrm{Bun}_G$.