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Topologies on split Kac-Moody groups over valued fields

Auguste Hebert

TL;DR

This work develops a topology-aware framework for split Kac-Moody groups $G$ over valued fields, grounding the construction in the masure geometry and parahoric data. It introduces two topologies: $\mathscr{T}_{\mathrm{Fix}}$, the coarsest topology making the Iwahori subgroup open, and a finer, Hausdorff topology $\mathscr{T}$ built from a regular filtration $(\mathcal{V}_{n\lambda})$, proving $\mathscr{T}$ refines $\mathscr{T}_{\mathrm{Fix}}$ and comparing both with the Kac-Peterson topology $\mathscr{T}_{KP}$. The paper shows $\mathscr{T}$ is strictly coarser than $\mathscr{T}_{KP}$ for non-reductive $G$, while retaining desired properties such as the openness of parahoric and Iwahori subgroups and the nonexistence of nontrivial compact interiors. An explicit affine $\mathrm{SL}_2$ example clarifies the filtrations, and the framework aligns with representation-theoretic goals by connecting to Hecke algebras and principal series representations. Overall, it provides a robust, geometry-informed topological structure on $G$ that supports both algebraic and analytic perspectives in the Kac-Moody setting.

Abstract

Let $G$ be a minimal split Kac-Moody group over a valued field {\mathcal{K}. Motivated by the representation theory of $G$, we define two topologies of topological group on $G$, which take into account the topology on {\mathcal{K}.

Topologies on split Kac-Moody groups over valued fields

TL;DR

This work develops a topology-aware framework for split Kac-Moody groups over valued fields, grounding the construction in the masure geometry and parahoric data. It introduces two topologies: , the coarsest topology making the Iwahori subgroup open, and a finer, Hausdorff topology built from a regular filtration , proving refines and comparing both with the Kac-Peterson topology . The paper shows is strictly coarser than for non-reductive , while retaining desired properties such as the openness of parahoric and Iwahori subgroups and the nonexistence of nontrivial compact interiors. An explicit affine example clarifies the filtrations, and the framework aligns with representation-theoretic goals by connecting to Hecke algebras and principal series representations. Overall, it provides a robust, geometry-informed topological structure on that supports both algebraic and analytic perspectives in the Kac-Moody setting.

Abstract

Let be a minimal split Kac-Moody group over a valued field {\mathcal{K}. Motivated by the representation theory of , we define two topologies of topological group on , which take into account the topology on {\mathcal{K}.
Paper Structure (33 sections, 43 theorems, 87 equations)

This paper contains 33 sections, 43 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Motivation from representation theory
  3. Topology on $G$, masure and main results
  4. Kac-Moody groups and masures
  5. Standard apartment of a masure
  6. Root generating system
  7. Vectorial apartment
  8. Split Kac-Moody groups over fields
  9. Minimal Kac-Moody groups over fields
  10. The affine group scheme $\mathfrak{U}^{pma}$
  11. Mathieu's group $\mathfrak{G}^{pma}$
  12. Minimal Kac-Moody group over rings
  13. Split Kac-Moody groups over valued fields and masures
  14. Action of $N$ on $\mathbb{A}$
  15. Affine apartment
  16. ...and 18 more sections

Key Result

Theorem 1.1

(see Theorem t_conjugation_invariance, Lemma l_decomposition_Vlambda, Proposition p_comparison_T_TFix and Proposition p_nonCompactness):

Theorems & Definitions (90)

  • Theorem 1.1
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Proposition 2.5
  • proof
  • Remark 2.6
  • ...and 80 more