Imaginary cone and reflection subgroups of Coxeter groups
The imaginary cone of a Kac–Moody Lie algebra is the convex hull of zero and the positive imaginary roots. This paper studies a related notion of 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 convex cone contained in the pointed convex 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. Finally, it is shown in general that the imaginary cone is the pointed convex cone spanned by the union of the imaginary cones of dihedral reflection subgroups.