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

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 $\mathcal{G}$.