Root group data (RGD) systems of affine type for significant subgroups of isotropic reductive groups over $k[t,t^{-1}]$

TL;DR

This paper constructs root group data systems of affine type for significant subgroups of isotropic reductive groups over Laurent polynomial rings, extending to the entire group under certain conditions, using relative pinning maps and affine root properties.

Contribution

It introduces a method to build affine RGD systems for subgroups of isotropic reductive groups over $k[t,t^{-1}]$, extending to the whole group with additional assumptions.

Findings

Constructed affine RGD systems for subgroups of $ ext{G}(k[t,t^{-1}])$

Extended RGD systems to entire groups under specific conditions

Utilized properties of affine root groups and reflections for verification

Abstract

Given a connected isotropic reductive not necessarily split $k$-group $\mathcal{G}$ with irreducible relative root system, we construct root group data (RGD) system of affine type for significant subgroups of $\mathcal{G}(k[t,t^{-1}])$, which can be extended to the whole group $\mathcal{G}(k[t,t^{-1}])$ under certain additional requirements. We rely on the relative pinning maps from paper "Elementary subgroups of isotropic reductive groups" by V. Petrov and A. Stavrova to construct the affine root groups. To verify the RGD axioms, we utilize the properties of the affine root groups, and the properties of reflections associated with the $k$-roots of $\mathcal{G}$.