The affine Grassmannian as a presheaf quotient
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912380218245120 |
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| author | Cesnavicius, Kestutis |
| author_facet | Cesnavicius, Kestutis |
| contents | 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 $\mathbb{P}^1_A$ that is ultimately built on the geometry of $\mathrm{Bun}_G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_04314 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The affine Grassmannian as a presheaf quotient Cesnavicius, Kestutis Algebraic Geometry Primary 14L15, Secondary 14M17 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 $\mathbb{P}^1_A$ that is ultimately built on the geometry of $\mathrm{Bun}_G$. |
| title | The affine Grassmannian as a presheaf quotient |
| topic | Algebraic Geometry Primary 14L15, Secondary 14M17 |
| url | https://arxiv.org/abs/2401.04314 |