Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Towards the tropicalization of reductive groups

Desmond Coles, Martin Ulirsch

TL;DR

This work develops a two-pronged framework for tropicalizing a connected reductive group $G$ using (i) Berkovich geometry and extended affine buildings to define toroidal bordifications and skeleta, and (ii) spherical tropicalization linked to wonderful compactifications and moduli of equivariant bundles. It builds a coherent apparatus of stacky cones, stacky fans, and toroidal bordifications to produce tropicalization maps that admit deformation retractions onto toroidal skeleta and are compatible with spherical tropicalization. The results reveal a tight correspondence between extended buildings, toroidal skeleta, and wonderful compactifications, with functorial behavior under group homomorphisms and a moduli-theoretic interpretation via bundles on chains of projective lines. The framework unifies non-Archimedean analytic, toric/toroidal, and moduli-theoretic perspectives on tropicalizations of reductive groups, enabling precise comparisons across toroidal and spherical settings and informing applications to spherical varieties and bundle moduli.

Abstract

Let $G$ be a connected reductive algebraic group over an algebraically closed field of characteristic zero carrying the trivial valuation. In this article we discuss two candidates for what could be the tropicalization of $G$. Our first suggestion is the extended affine building associated to $G$. This perspective makes makes use of Berkovich's embedding of the extended affine building into the Berkovich analytic space $G^{\textrm{an}}$ and expands on work of Mumford by associating a toroidal bordification of $G$ to the choice of stacky fan in the building. We show that the natural retraction onto the building is compatible with the tropicalization map associated to a toroidal bordification. Our second suggestion is a Weyl chamber of $G$, a special instance of spherical tropicalization, where we think of $G$ as a spherical $G\times G$-variety with respect to left-right-multiplication. We show that the spherical tropicalization map may be identified with the toroidal tropicalization map associated to a wonderful compactification of $G$. This map also has a moduli-theoretic interpretation expanding on the compactifications of $G$ as moduli spaces of framed $\mathbb{G}_m$-equivariant principal bundles on chains of projective lines introduced by Martens and Thaddeus.

Towards the tropicalization of reductive groups

TL;DR

This work develops a two-pronged framework for tropicalizing a connected reductive group using (i) Berkovich geometry and extended affine buildings to define toroidal bordifications and skeleta, and (ii) spherical tropicalization linked to wonderful compactifications and moduli of equivariant bundles. It builds a coherent apparatus of stacky cones, stacky fans, and toroidal bordifications to produce tropicalization maps that admit deformation retractions onto toroidal skeleta and are compatible with spherical tropicalization. The results reveal a tight correspondence between extended buildings, toroidal skeleta, and wonderful compactifications, with functorial behavior under group homomorphisms and a moduli-theoretic interpretation via bundles on chains of projective lines. The framework unifies non-Archimedean analytic, toric/toroidal, and moduli-theoretic perspectives on tropicalizations of reductive groups, enabling precise comparisons across toroidal and spherical settings and informing applications to spherical varieties and bundle moduli.

Abstract

Let be a connected reductive algebraic group over an algebraically closed field of characteristic zero carrying the trivial valuation. In this article we discuss two candidates for what could be the tropicalization of . Our first suggestion is the extended affine building associated to . This perspective makes makes use of Berkovich's embedding of the extended affine building into the Berkovich analytic space and expands on work of Mumford by associating a toroidal bordification of to the choice of stacky fan in the building. We show that the natural retraction onto the building is compatible with the tropicalization map associated to a toroidal bordification. Our second suggestion is a Weyl chamber of , a special instance of spherical tropicalization, where we think of as a spherical -variety with respect to left-right-multiplication. We show that the spherical tropicalization map may be identified with the toroidal tropicalization map associated to a wonderful compactification of . This map also has a moduli-theoretic interpretation expanding on the compactifications of as moduli spaces of framed -equivariant principal bundles on chains of projective lines introduced by Martens and Thaddeus.

Paper Structure

This paper contains 36 sections, 31 theorems, 93 equations, 2 figures.

Table of Contents

  1. Berkovich spaces and extended affine buildings
  2. Berkovich analytic geometry
  3. Analytification
  4. The $(.)^\beth$ analytification
  5. The Berkovich spectrum
  6. Berkovich's $\ast$-multiplication
  7. Reductive groups and extended buildings
  8. Structure and combinatorics of reductive groups
  9. Bruhat-Tits buildings
  10. Toroidal bordifications and tropicalization
  11. Toric varieties, toroidal embeddings, and tropicalization
  12. Toric geometry and tropicalization
  13. Rational polyhedral cones and affine toric varieties
  14. Tropicalization of a toric variety
  15. Toroidal embeddings and their tropicalization
  16. ...and 21 more sections

Key Result

Theorem 1

Let $G$ be a reductive group. For every stacky rational polyhedral fan $(\Delta,\Phi)$ in $G^{\mathop{\mathrm{}}\nolimits}$ there is a separated Deligne-Mumford stack $\mathcal{X}_G(\Delta)$ locally of finite type over $k$ that contains $G$ as a dense open subset such that the following properties h

Figures (2)

  • Figure 1: The lattice $N$ and one Weyl chamber for $\mathop{\mathrm{PGL}}\nolimits(3)$ (on the left) and $\mathop{\mathrm{SL}}\nolimits(3)$ (on the right). Notice that the Weyl chamber for $\mathop{\mathrm{PGL}}\nolimits(3)$ is a unimodular cone, i.e. the spanning vectors form a basis for the lattice, but the Weyl chamber for $\mathop{\mathrm{SL}}\nolimits(3)$ is not unimodular.
  • Figure 2: An example of a fan $\Delta$ and its canonical compactification $\overline{\Delta}$.

Theorems & Definitions (88)

  • Remark
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Corollary 5
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Proposition 1.5
  • ...and 78 more