Affine Grassmannians in A^1-algebraic topology

TL;DR

This paper establishes a canonical motivic equivalence between Gm-loops of a reductive group G and its affine Grassmannian, and computes the motive of the loop space in the motivic derived category.

Contribution

It proves a canonical motivic equivalence for isotropic reductive groups and computes the motive of the Gm-loop space in the motivic setting.

Findings

Established a motivic equivalence Omega_Gm G = Gr_G for isotropic reductive G.

Computed the motive M(Omega_Gm G) in DM(k, Z) for perfect fields.

Provided new tools for understanding affine Grassmannians in A^1-algebraic topology.

Abstract

Let k be a field. Denote by Spc(k)_* the unstable, pointed motivic homotopy category and by Omega_Gm: Spc(k)_* \to Spc(k)_* the Gm-loops functor. For a k-group G, denote by Gr_G the affine Grassmannian of G. If G is isotropic reductive, we provide a canonical motivic equivalence Omega_Gm G = Gr_G. If k is perfect, we use this to compute the motive M(Omega_Gm G) in DM(k, Z).