Residues on Affine Grassmannians

TL;DR

This paper introduces a new concept of index and residue for elements of linear groups over formal Laurent series fields, providing an alternative proof of Gabber's theorem and linking to affine Grassmannian theory.

Contribution

It defines residues and indices for elements of linear groups over k((t)), offering a novel proof of Gabber's theorem and connecting to affine Grassmannians for reductive groups.

Findings

Provided an alternative proof of Gabber's theorem.

Established a connection between residues and affine Grassmannians.

Characterized subgroups of linear groups over formal Laurent series.

Abstract

Given a linear group G over a field k, we define a notion of index and residue of an element g of G(k((t)). This provides an alternative proof of Gabber's theorem stating that G has no subgroups isomorphic to the additive or the commutative group iff G(k[[t]])= G(k((t))). In the case of a reductive group, we offer an explicit connection with the theory of affine grassmannians.