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 étale 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 $G$ 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 ${\mathbb{P}}_A^1$ that is ultimately built on the geometry of ${\mathrm{Bun}}_G$.

arXiv comment11 pages; final version, to appear in Comptes Rendus Mathématique, Académie des Sciences, Paris; updated the numbering scheme to match the published version