Limit directions for Lorentzian Coxeter systems

Abstract

Every Coxeter group admits a geometric representation as a group generated by reflections in a real vector space. In the projective representation space of a Coxeter group, limit directions arising from a point are accumulation points of the orbit of this point. In particular, limit directions arising from roots are called limit roots. Recent studies show that limit roots lie on the isotropic cone of the representation space. In this paper, we study limit directions of Coxeter groups arising from any point when the representation space is a Lorentz space. We prove that the limit roots are the only light-like limit directions, and characterize the limit roots using eigenvectors of infinite-order elements. Then we describe the structure of space-like limit directions in terms of the projective Coxeter arrangement. Some non-Lorentzian cases are also discussed.