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Relative Bruhat decomposition of wonderful compactification

Fei Chen, Shang Li

TL;DR

The paper develops a comprehensive relative Bruhat theory for the wonderful compactification of a reductive group over a general base field. It constructs a precise orbit stratification of \\overline{G}(k) into P(k)×P^−(k)-orbits, refined from G(k)×G(k)-orbits, and provides a geometric refinement and an explicit relative Bruhat order describing when one orbit lies in the closure of another. The results extend Brion–Springer-type theorems from algebraically closed fields to arbitrary base fields via Galois descent, while also addressing topological closure properties under non-discrete topologies. Together, these findings illuminate the asymptotic (infinity) behavior of Bruhat decompositions in the setting of wonderful compactifications and unify geometric and combinatorial descriptions across split and non-split groups. The framework has potential implications for representation theory, invariant theory, and the study of compactifications of reductive groups over general fields.

Abstract

In the seminal paper of Borel and Tits about reductive groups, they show some fundamental results about Bruhat cells with respect to a minimal parabolic subgroup, e.g., relative Bruhat decomposition and its geometrization, relative Bruhat order and the relation of Zariski closure and topological closure. In this paper, we show analogous results for Bruhat cells of wonderful group compactification in the sense of De Concini and Procesi. Our results can be viewed as the version at infinity of those of Borel and Tits. Our main focus is general base field. When the base field is algebraically closed, most of our results are proved by Brion and Springer.

Relative Bruhat decomposition of wonderful compactification

TL;DR

The paper develops a comprehensive relative Bruhat theory for the wonderful compactification of a reductive group over a general base field. It constructs a precise orbit stratification of \\overline{G}(k) into P(k)×P^−(k)-orbits, refined from G(k)×G(k)-orbits, and provides a geometric refinement and an explicit relative Bruhat order describing when one orbit lies in the closure of another. The results extend Brion–Springer-type theorems from algebraically closed fields to arbitrary base fields via Galois descent, while also addressing topological closure properties under non-discrete topologies. Together, these findings illuminate the asymptotic (infinity) behavior of Bruhat decompositions in the setting of wonderful compactifications and unify geometric and combinatorial descriptions across split and non-split groups. The framework has potential implications for representation theory, invariant theory, and the study of compactifications of reductive groups over general fields.

Abstract

In the seminal paper of Borel and Tits about reductive groups, they show some fundamental results about Bruhat cells with respect to a minimal parabolic subgroup, e.g., relative Bruhat decomposition and its geometrization, relative Bruhat order and the relation of Zariski closure and topological closure. In this paper, we show analogous results for Bruhat cells of wonderful group compactification in the sense of De Concini and Procesi. Our results can be viewed as the version at infinity of those of Borel and Tits. Our main focus is general base field. When the base field is algebraically closed, most of our results are proved by Brion and Springer.

Paper Structure

This paper contains 22 sections, 27 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Conventions and notation
  3. Acknowledgments
  4. Wonderful compactification
  5. Group setup
  6. Absolute root system
  7. Relative root system
  8. $\ast$-action
  9. Split case
  10. Construction
  11. Properties of $\overline{G_{k_s}}$
  12. Galois descent
  13. Relative Bruhat decomposition
  14. $G(k)\times G(k)$-orbits
  15. $P(k)\times P^-(k)$-orbits
  16. ...and 7 more sections

Key Result

Theorem 1.1

(boreltits, boreltitscomplement) (1). (Relative Bruhat decomposition) $G(k)$ is the disjoint union of $P(k)\cdot w\cdot P(k)$ with $w\in W$. (2). (Geometric refinement of (1)) If $P\cdot w\cdot P$ meets $P\cdot w'\cdot P$ inside $G$, then $w=w'$. (3). (Relative Bruhat order) If $k$ is infinite, we h (4). (Topological closure) Suppose that $k$ is a topological field equipped with a non-discrete top

Theorems & Definitions (50)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 40 more