A causal characterisation of $\mathrm{Sp}_{\mathrm{ell}}^{+}(2n)$

TL;DR

This paper demonstrates that the conjugation invariant cone structure on the positive elliptic region of the symplectic group is globally hyperbolic, providing insights into Lorentzian distances and causality within this mathematical setting.

Contribution

It establishes the global hyperbolicity of the cone structure on the positively elliptic region of the symplectic group and characterizes this region and specific elements via causality.

Findings

Proves global hyperbolicity of the cone structure in the elliptic region.

Provides a formula for a bi-invariant Lorentzian distance function.

Characterizes the elliptic region and -id in terms of causality.

Abstract

We show that the natural conjugation invariant cone structure on the linear symplectic group $\mathrm{Sp}(2n)$ is globally hyperbolic in the positively elliptic region $\mathrm{Sp}_{\mathrm{ell}}^{+}(2n)$. This answers a question by Abbondandolo, Benedetti and Polterovich and shows a formula for a bi-invariant Lorentzian distance function dened by these authors for elements in this region. Moreover we give a characterisation of the positively elliptic region and of $-id \in\mathrm{Sp}(2n)$ in terms of the causality of this cone structure.