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Elliptic domains in Lie groups

Jakob Hedicke, Karl-Hermann Neeb

Abstract

An element $g$ of a Lie group is called stably elliptic if it is contained in the interior of the set $G^e$ of elliptic elements, characterized by the property that $\mathrm{Ad}(g)$ generates a relatively compact subgroup. Stably elliptic elements appear naturally in the geometry of causal symmetric spaces and in representation theory. We characterize stably elliptic elements in terms of the fixed point algebra of $\mathrm{Ad}(g)$ and show that the connected components of the set $G^{se}$ of stably elliptic elements can be described in terms of the Weyl group action on a compactly embedded Cartan subalgebra. In the case of simple hermitian Lie groups we relate stably elliptic elements to maximal invariant cones and the associated subsemigroups. In particular we show that the basic connected component $G^{se}(0)$ can be characterized in terms of the compactness of order intervals and that $G^{se}(0)$ is globally hyperbolic with respect to the induced biinvariant causal structure.

Elliptic domains in Lie groups

Abstract

An element of a Lie group is called stably elliptic if it is contained in the interior of the set of elliptic elements, characterized by the property that generates a relatively compact subgroup. Stably elliptic elements appear naturally in the geometry of causal symmetric spaces and in representation theory. We characterize stably elliptic elements in terms of the fixed point algebra of and show that the connected components of the set of stably elliptic elements can be described in terms of the Weyl group action on a compactly embedded Cartan subalgebra. In the case of simple hermitian Lie groups we relate stably elliptic elements to maximal invariant cones and the associated subsemigroups. In particular we show that the basic connected component can be characterized in terms of the compactness of order intervals and that is globally hyperbolic with respect to the induced biinvariant causal structure.

Paper Structure

This paper contains 19 sections, 39 theorems, 227 equations.

Table of Contents

  1. Introduction
  2. Elliptic elements of Lie groups
  3. Existence of stably elliptic elements
  4. Charactization of stably elliptic elements
  5. Polar decomposition of elliptic domains
  6. Another characterization of compact elements
  7. Connected components of $G^{se}$
  8. Root decomposition
  9. Specification of connected components
  10. Classification of connected components
  11. The case of simple hermitian Lie algebras
  12. Elliptic domains and compact order intervals in semigroups
  13. The maximal invariant cone and ${\mathfrak g}^{se}(0)$
  14. The Guichardet--Wigner quasimorphism
  15. The causal flag manifold
  16. ...and 4 more sections

Key Result

Lemma 2.3

Let $K \subseteq G$ be the integral subgroup corresponding to a maximal compactly embedded subalgebra ${\mathfrak k} \subseteq {\mathfrak g}$. Then the following assertions hold:

Theorems & Definitions (102)

  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 92 more