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Hurwitz-Radon numbers and proper actions of semisimple Lie groups

Kazuki Kannaka, Koichi Tojo

TL;DR

This work classifies proper isometric actions of non-compact semisimple Lie groups on pseudo-Riemannian symmetric spaces by introducing Property (VE), which captures the boundary between existence and non-existence of such actions. A central technique is reformulating the Hurwitz–Radon number via Lie algebra representations, yielding ρ^{(i)}(𝔤,ι) that bound Spin(n,1) actions on VE spaces; in particular, Spin(n,1) actions on the boundary family \\mathbf{H}_{+}^{N,N-1} exist precisely for 2 ≤ n ≤ ρ(N). The authors develop a comprehensive framework connecting Clifford algebras, spin representations, and embedding criteria to compute these HR-type numbers for classical pairs, and they classify VE symmetric spaces with proper SL(2,R) actions, including non-symmetric VE examples. The results have implications for rigidity phenomena in pseudo-Riemannian geometry and the study of Clifford–Klein forms, linking representation theory with geometric properness in indefinite settings.

Abstract

We study proper isometric actions of non-compact semisimple Lie groups on symmetric spaces in pseudo-Riemannian geometry. Motivated by Okuda's classification of semisimple symmetric spaces admitting proper $SL(2,\mathbb{R})$-actions [J. Differ. Geom., 2013], we focus on a family of symmetric spaces lying on the "boundary case" of the existence of proper $SL(2,\mathbb{R})$-actions. As a rigidity result, we show that any connected non-compact semisimple Lie group acting properly on this family must be globally isomorphic to $Spin(n,1)$ up to compact factors. Moreover, the largest value of $n$ for the existence of $Spin(n,1)$-proper actions is governed by the Hurwitz-Radon number. Our family includes the pseudo-Riemannian hyperbolic space $\mathbf{H}_{+}^{N,N-1}$ of signature $(N,N-1)$.

Hurwitz-Radon numbers and proper actions of semisimple Lie groups

TL;DR

This work classifies proper isometric actions of non-compact semisimple Lie groups on pseudo-Riemannian symmetric spaces by introducing Property (VE), which captures the boundary between existence and non-existence of such actions. A central technique is reformulating the Hurwitz–Radon number via Lie algebra representations, yielding ρ^{(i)}(𝔤,ι) that bound Spin(n,1) actions on VE spaces; in particular, Spin(n,1) actions on the boundary family \\mathbf{H}_{+}^{N,N-1} exist precisely for 2 ≤ n ≤ ρ(N). The authors develop a comprehensive framework connecting Clifford algebras, spin representations, and embedding criteria to compute these HR-type numbers for classical pairs, and they classify VE symmetric spaces with proper SL(2,R) actions, including non-symmetric VE examples. The results have implications for rigidity phenomena in pseudo-Riemannian geometry and the study of Clifford–Klein forms, linking representation theory with geometric properness in indefinite settings.

Abstract

We study proper isometric actions of non-compact semisimple Lie groups on symmetric spaces in pseudo-Riemannian geometry. Motivated by Okuda's classification of semisimple symmetric spaces admitting proper -actions [J. Differ. Geom., 2013], we focus on a family of symmetric spaces lying on the "boundary case" of the existence of proper -actions. As a rigidity result, we show that any connected non-compact semisimple Lie group acting properly on this family must be globally isomorphic to up to compact factors. Moreover, the largest value of for the existence of -proper actions is governed by the Hurwitz-Radon number. Our family includes the pseudo-Riemannian hyperbolic space of signature .
Paper Structure (35 sections, 64 theorems, 141 equations, 1 figure, 10 tables)

This paper contains 35 sections, 64 theorems, 141 equations, 1 figure, 10 tables.

Table of Contents

  1. Introduction
  2. Illustrative example of main results
  3. Background and Motivation
  4. Overview of the main theorems
  5. Organization of the paper
  6. Notation and Conventions
  7. Main results and ideas of the proofs
  8. Preliminary: Kobayashi's properness criterion
  9. Property (VE) for homogeneous spaces
  10. Reformulation of the Hurwitz-Radon number in terms of Lie algebra representations
  11. Hurwitz-Radon number and proper actions
  12. Questions
  13. Semisimple Lie groups that never act properly
  14. Characterization of the orthogonal Lie algebra of real rank one
  15. Proof of Theorem C
  16. ...and 20 more sections

Key Result

Theorem 1.1

Let $L$ be a connected semisimple Lie group with no compact factors. Then $\mathbf{H}_{+}^{N,N-1}$ admits a proper isometric action of $L$ if and only if $L$ is globally isomorphic to $Spin(n,1)$ with $2\leq n\leq \rho(N)$.

Figures (1)

  • Figure 1: $\mathbf{H}_{+}^{p,q}$ and proper $SL(2,\mathbb{R})$-actions.

Theorems & Definitions (126)

  • Theorem 1.1: See Section \ref{['section:intro-classification-proper']} for the proof
  • Corollary 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • Lemma 2.8
  • proof
  • Theorem 1
  • Remark 2.9
  • ...and 116 more