nLab affine Grassmannian

Idea

According to wikipedia,

the affine Grassmannian of an algebraic group $G$ over a field $k$ is an ind-scheme which can be thought of as a flag variety for the loop group $G(k((t)))$

The affine Grassmannian is ind-representable. Affine Grassmannian of $SL_n$ admits embedding into Sato Grassmanian.

Definitions

The $\GL_{n}$ case

Definition

(Definition 1.1.1 of #Zhu2016) Let $k$ be a field and let $R$ be a $k$-algebra. An $R$-family of lattices in $k((t))^{n}$ is a finitely generated projective submodule $\Lambda$ of $R((t))^{n}$ such that $\Lambda \otimes_{R[[t]]}R((t))=R((t))^{n}$.

Definition

(Definition 1.1.2 of #Zhu2016) The affine Grassmannian $\Gr_{\GL_{n}}$ for $\GL_{n}$ is the presheaf that assigns to every $k$-algebra $R$ the set of $R$-families of lattices in $k((t))^{n}$.

The general case

Definition

(Section 1.2 of #Zhu2016) Let $k$ be a field and let $G$ be an affine $k$-group. If $R$ is a $k$-algebra, let $D_{R}=\Spec k[[t]]\widehat{\times}\Spec R$ and let $D^{*}_{R}=\Spec k((t))\widehat{\times} \Spec R$. Let $\mathcal{E}^{0}$ be the trivial $G$-torsor over $D_{R}$.

The affine Grassmannian $\Gr_{G}$ of $G$ is the presheaf that assigns to every $k$-algebra $R$ the set of pairs $(\mathcal{E},\beta)$, where $\mathcal{E}$ is a $G$-torsor on $D_{R}$ and $\beta:\mathcal{E}\vert_{D^{*}_{R}}\xrightarrow{\simeq}\mathcal{E}^{0}\vert_{D^{*}_{R}}$ is a trivialization.

Related

• geometric Langlands correspondence, Ran space
• G-torsor?
• Grassmannian, flag variety

References

• Evgeny Feigin, Michael Finkelberg, Markus Reineke, Degenerate affine Grassmannians and loop quivers, http://arxiv.org/abs/1410.0777

• Xinwen Zhu, An introduction to affine Grassmannians and the geometric Satake equivalence, arXiv:1603.05593.

• Bhargav Bhatt, Peter Scholze, Projectivity of the Witt vector affine Grassmannian, Invent. math. 209, 329-423 (2017) doi arXiv:1507.06490

• Alexander Schmitt, Affine flag manifolds and principal bundles, Trends in Mathematics, Springer 2010

category: algebra, algebraic geometry