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

Abstract

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