[Paper Review] Reductive group actions

Friedrich Knop, Bernhard Krötz|arXiv (Cornell University)|Apr 4, 2016

Advanced Algebra and GeometryMathematics33 references21 citations

TL;DR

This paper develops a unified theory of reductive group actions over arbitrary fields of characteristic zero by introducing a k-version of the local structure theorem and defining a generalized Tits index that combines Luna’s spherical systems with Borel-Tits theory. The key contribution is a k-spherical root system and a canonical compactification of k-rational points in k-spherical varieties, generalizing the maximal Satake compactification for real groups.

ABSTRACT

In this paper, we study rationality properties of reductive group actions which are defined over an arbitrary field of characteristic zero. Thereby, we unify Luna's theory of spherical systems and Borel-Tits' theory of reductive groups. In particular, we define for any reductive group action a generalized Tits index whose main constituents are a root system and a generalization of the anisotropic kernel. The index controls to a large extent the behavior at infinity (i.e., embeddings). For k-spherical varieties (i.e., where a minimal parabolic has an open orbit) we obtain explicit (wonderful) completions of the set of rational points. For local fields this means honest compactifications generalizing the maximal Satake compactification of a symmetric space. Our main tool is a k-version of the local structure theorem.

Motivation & Objective

• To unify Luna’s theory of spherical varieties with Borel-Tits theory of reductive groups over non-closed fields.
• To develop a rationality theory for reductive group actions over arbitrary fields of characteristic zero, especially for non-split or non-quasi-split groups.
• To define a generalized Tits index that controls the asymptotic behavior of embeddings and captures the structure of k-rational points at infinity.
• To construct explicit compactifications of k-rational points in k-spherical varieties, generalizing the maximal Satake compactification.
• To establish a k-spherical root system and a k-anisotropic kernel that generalize the classical root system and anisotropic kernel in the algebraically closed case.

Proposed method

• Introduce a k-version of the local structure theorem, decomposing a G-variety X into an elementary kernel X_el and an anisotropic kernel X_an via quotients by N and AN.
• Define the k-rank of X as rk_k X = dim A_k(X), where A_k(X) is a quotient torus of the maximal split torus A.
• Introduce k-central valuations—G-invariant valuations trivial on AN-invariants—that classify irreducible G-stable divisors in birational models of X.
• Construct a Weyl group W_k(X) acting on the character lattice Ξ_k(X), leading to a Weyl chamber in the rational vector space Hom(Ξ_k(X), ℚ).
• Define the k-spherical roots Σ_k(X) as a set of simple roots for a root system Φ_k(X) ⊂ Ξ_k(X), which is the restricted root system of the K-spherical root system Φ_K(X) to the maximal split torus A.
• Use the fan of the generalized Tits index to construct a compactification X_st(ℝ) of X(ℝ) for k=ℝ, which is a compact Σ-manifold equivariant under G(ℝ)^0.

Experimental results

Research questions

• RQ1 How can Luna’s theory of spherical varieties be extended to non-split reductive groups over arbitrary fields of characteristic zero?
• RQ2 What is the correct generalization of the Tits index for reductive groups over non-closed fields, incorporating rationality and asymptotic behavior?
• RQ3 How do k-rational points behave at infinity in spherical varieties, and can they be compactified in a canonical way?
• RQ4 What is the relationship between the k-spherical root system and the classical spherical root system over the algebraic closure?
• RQ5 Can the maximal Satake compactification of symmetric spaces be generalized to arbitrary k-spherical varieties over local fields?

Key findings

• The k-spherical roots Σ_k(X) form a root system Φ_k(X) in the character lattice Ξ_k(X), which is the restricted root system of the K-spherical root system Φ_K(X) to the maximal split torus A.
• For k=ℝ, the compactification X_st(ℝ) of X(ℝ) is a compact Σ-manifold equivariant under G(ℝ)^0, and its boundary strata are precisely the G(ℝ)^0-orbits.
• When X is k-wonderful, the fan used in the compactification can be taken as the standard fan, and the resulting compactification realizes the maximal Satake compactification for symmetric spaces.
• In the case of real rank 1, the compactification X_st(ℝ) is homeomorphic to S^2, a 2-torus, a real projective plane, or a Klein bottle, depending on the spherical homogeneous space G/H.
• The boundary Y(ℝ) of X_st(ℝ) separates the real locus into components corresponding to different G(ℝ)^0-orbits, and cutting along Y(ℝ) yields disjoint unions of disks or Möbius strips, reflecting the orbit structure.
• For horospherical varieties, the compactification may be non-orientable (e.g., a Klein bottle), and cutting along closed orbits yields non-trivial covers, indicating non-trivial topology of the real locus.