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

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.

10 pages