nLab affine Grassmannian

Contents

Contents

Idea

According to wikipedia,

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

The affine Grassmannian is ind-representable. Affine Grassmannian of SL nSL_n admits embedding into Sato Grassmanian.

Definitions

The GL n\GL_{n} case

Definition

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

Definition

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

The general case

Definition

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

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

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

Last revised on August 15, 2023 at 20:23:02. See the history of this page for a list of all contributions to it.