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Twin masures associated with Kac-Moody groups over Laurent polynomials

Nicole Bardy-Panse, Auguste Hebert, Guy Rousseau

TL;DR

The article extends the Masure/Kac–Moody framework to study actions of split Kac–Moody groups over Laurent polynomial fields on twin masures, connecting these actions to a Kazhdan–Lusztig–style theory. It develops Bruhat and Iwasawa-type decompositions for dense subrings, introduces a conjectural well-defined retraction ${\rho}_{C_\infty}$ in the twin setting, and analyzes $C_{\infty}$-Hecke paths and their liftings, providing finite-count formulas in many cases. A key contribution is the construction of a twin-masure apparatus to relate line segments, retractions, and Hecke paths, with explicit treatment of the affine ${\rm SL}_2$ case showing that full Birkhoff decompositions fail in the twin group, thereby delineating the limits of current approaches. Overall, the work lays groundwork for a Kac–Moody KL theory in twin masures, yielding new tools for counting liftings and connecting group-theoretic decompositions to geometric retractions. The results have potential implications for representation-theoretic invariants and combinatorial aspects of Kac–Moody groups over valued/function fields.

Abstract

Let $\mathfrak{G}$ be a split reductive group, $\mathbb{k}$ be a field and $\varpi$ be an indeterminate. In order to study $\mathfrak{G}(\mathbb{k}[\varpi,\varpi^{-1}])$ and $\mathfrak{G}(\mathbb{k}(\varpi))$, one can make them act on their twin building $\mathcal{I} = \mathcal{I}_\oplus\times \mathcal{I}_\ominus$, where $\mathcal{I}_\oplus$ and $\mathcal{I}_\ominus$ are related via a ''codistance''. Masures are generalizations of Bruhat-Tits buildings adapted to the study of Kac-Moody groups over valued fields. Motivated by the work of Dinakar Muthiah on Kazhdan-Lusztig polynomials associated with Kac-Moody groups, we study the action of $\mathfrak{G}(\mathbb{k}[\varpi,\varpi^{-1}])$ and $\mathfrak{G}(\mathbb{k}(\varpi,\varpi^{-1}))$ on their ''twin masure'', when $\mathfrak{G}$ is a split Kac-Moody group instead of a reductive group.

Twin masures associated with Kac-Moody groups over Laurent polynomials

TL;DR

The article extends the Masure/Kac–Moody framework to study actions of split Kac–Moody groups over Laurent polynomial fields on twin masures, connecting these actions to a Kazhdan–Lusztig–style theory. It develops Bruhat and Iwasawa-type decompositions for dense subrings, introduces a conjectural well-defined retraction in the twin setting, and analyzes -Hecke paths and their liftings, providing finite-count formulas in many cases. A key contribution is the construction of a twin-masure apparatus to relate line segments, retractions, and Hecke paths, with explicit treatment of the affine case showing that full Birkhoff decompositions fail in the twin group, thereby delineating the limits of current approaches. Overall, the work lays groundwork for a Kac–Moody KL theory in twin masures, yielding new tools for counting liftings and connecting group-theoretic decompositions to geometric retractions. The results have potential implications for representation-theoretic invariants and combinatorial aspects of Kac–Moody groups over valued/function fields.

Abstract

Let be a split reductive group, be a field and be an indeterminate. In order to study and , one can make them act on their twin building , where and are related via a ''codistance''. Masures are generalizations of Bruhat-Tits buildings adapted to the study of Kac-Moody groups over valued fields. Motivated by the work of Dinakar Muthiah on Kazhdan-Lusztig polynomials associated with Kac-Moody groups, we study the action of and on their ''twin masure'', when is a split Kac-Moody group instead of a reductive group.
Paper Structure (72 sections, 54 theorems, 48 equations, 2 figures)

This paper contains 72 sections, 54 theorems, 48 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Context
  3. Split reductive groups over valued fields and Bruhat-Tits buildings
  4. Twin building of $\mathfrak{G}({\Bbbk}[\varpi,\varpi^{-1}])$
  5. Kazhdan-Lusztig polynomials
  6. Split Kac-Moody groups over valued field and masures
  7. Kazhdan-Lusztig polynomials in the Kac-Moody setting
  8. Content of this paper
  9. Split Kac-Moody groups over valued fields and masures
  10. Standard apartment of a masure
  11. Root generating system
  12. Local chambers, sectors, chimneys
  13. Split Kac-Moody groups over valued fields
  14. Minimal split Kac-Moody groups
  15. Subgroups $N$ and $N_{\mathcal{R}}$
  16. ...and 57 more sections

Key Result

Theorem 1

We have

Figures (2)

  • Figure 1: $C_\infty$-Hecke path
  • Figure :

Theorems & Definitions (118)

  • Theorem
  • Remark 2.1
  • Remark 2.2
  • Proposition
  • proof
  • Lemma
  • proof
  • Proposition
  • proof
  • Proposition
  • ...and 108 more