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

Yuan Zhang

TL;DR

The paper addresses constructing an explicit root group data (RGD) system of affine type for significant subgroups of isotropic reductive $k$-groups over $k[t,t^{-1}]$, including cases not split over $k$. It leverages Petrov–Stavrova relative pinning maps to define affine root groups and verifies RGD axioms, enabling an extension from a split-like elementary subgroup to the full group under additional hypotheses. The main contribution is an explicit, verifiable RGD framework for $igl\mathcal{G}(k[t,t^{-1}])^{+}C_{\mathcal{G}}(S)(k)\bigr)$ that yields affine twin buildings and closes gaps in the literature for non-split isotropic reductive groups. The results have potential applications in the study of group actions on twin buildings and in understanding Kac–Moody-like structures arising from non-split reductive groups over Laurent polynomial rings.

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 $\mathcal{G}$.

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

TL;DR

The paper addresses constructing an explicit root group data (RGD) system of affine type for significant subgroups of isotropic reductive -groups over , including cases not split over . It leverages Petrov–Stavrova relative pinning maps to define affine root groups and verifies RGD axioms, enabling an extension from a split-like elementary subgroup to the full group under additional hypotheses. The main contribution is an explicit, verifiable RGD framework for that yields affine twin buildings and closes gaps in the literature for non-split isotropic reductive groups. The results have potential applications in the study of group actions on twin buildings and in understanding Kac–Moody-like structures arising from non-split reductive groups over Laurent polynomial rings.

Abstract

Given a connected isotropic reductive not necessarily split -group with irreducible relative root system, we construct root group data (RGD) system of affine type for significant subgroups of , which can be extended to the whole group 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 -roots of .
Paper Structure (13 sections, 5 theorems, 33 equations)

This paper contains 13 sections, 5 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Motivation and history
  3. Summary
  4. Review on root group data (RGD) system and affine roots
  5. Root group data (RGD) systems
  6. Affine roots
  7. Review on reductive groups
  8. Summary of needed results
  9. Main theorem and its proof
  10. Preparations
  11. Main theorem
  12. Examples
  13. Acknowledgement

Key Result

Theorem 1

For a connected isotropic reductive affine algebraic group $\mathcal{G}$ with irreducible relative root system, where we denote by $\mathcal{G}(k[t,t^{-1}])^{+}$ the elementary subgroup (see definition of elementary subgroup at n:Important proteries of generalize unipotent elements and affine root g is an RGD system of affine type.

Theorems & Definitions (17)

  • Theorem
  • Remark
  • Definition 2.1: Prenilpotent pair of roots
  • Definition 2.2: General RGD system
  • Definition 3.1: Character
  • Definition 3.2: Cocharacter
  • Remark 3.3: Lie algebras of Chevalley group
  • Definition 3.4: Split reductive group and admissible isomorphism (i.e. absolute pinning isomorphism)
  • Definition 3.5: Reduced root system in a reductive group
  • Definition 3.6: $k$-root of reductive group
  • ...and 7 more