Reducedness of twisted loop groups

TL;DR

This work establishes an elementary proof of the reducedness of twisted loop groups $LG$ by adapting the Kneser–Tits strategy to the twisted setting. The approach reduces the problem to checking reducedness on Artinian local rings and centers on generating the maximal torus by unipotent subgroups via a detailed torus-generation analysis, including the SU$_3$ case. The primary results yield that $LG$ is reduced when $k$ is algebraically closed and $G$ is connected semisimple, simply connected, absolutely almost simple over $k((t))$, and extend to geometrical reducedness for general fields under coprimality conditions with $|\,\pi_1(G)|$. As a consequence, affine Grassmannians and affine flag varieties inherit geometric reducedness under the same hypotheses, strengthening the foundational understanding of the geometry of loop groups. The methods rely solely on standard algebraic group theory and group scheme techniques, with key steps addressing big cells, Artinian reduction, and orbit-wise Kneser–Tits arguments, providing a streamlined alternative to more analytic or condensed-mathematics approaches.

Abstract

We give an elementary proof of the reducedness of twisted loop groups along the lines of the Kneser-Tits problem.

Theorems & Definitions (69)

• Theorem 1: See \ref{['Theorem: LG is reduced']}
• Theorem 2
• Corollary 3
• Corollary 4
• Definition 5
• Definition 6
• Definition 7
• Definition 8
• Lemma 9
• Lemma 10