  The imaginary cone of a Kac-Moody Lie algebra is the convex hull of zero and
the positive imaginary roots. This paper studies the imaginary cone for a class
of root systems of general Coxeter groups W. It is shown that the imaginary
cone of a reflection subgroup of W is contained in that of W, and that for
irreducible infinite W of finite rank, the closed imaginary cone is the only
non-zero, closed, pointed W-stable cone contained in the pointed cone spanned
by the simple roots. For W of finite rank, various natural notions of faces of
the imaginary cone are shown to coincide, the face lattice is explicitly
described in terms of the lattice of facial reflection subgroups and it is
shown that the Tits cone and imaginary cone are related by a duality closely
analogous to the standard duality for polyhedral cones, even though neither of
them is a closed cone in general. Some of these results have application, to be
given in sequels to this paper, to dominance order of Coxeter groups,
associated automata, and construction of modules for generic Iwahori-Hecke
algebras.
