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 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. The imaginary cone ([30, Ch 5]) of a Kac-Moody Lie algebra is the convex hull of zero and the positive imaginary roots. The combinatorial characterization of imaginary roots from [30] can be used to give a definition ([26], [25]) of the imaginary cone which makes sense for possibly non-crystallographic root systems of Coxeter groups (which do not have imaginary roots in general). This paper systematically studies the imaginary cone for a class of root systems of general Coxeter groups W . As well as in [30] for the crystallographic case, some of the basic facts may be found in [25], [26], [22] and [27]. One of the main new results (Theorem 6.3) is that the imaginary cone of a reflection subgroup W ′ of W is contained in the imaginary cone of W ; the corresponding result for the closures of the imaginary cones is easier to prove but much less useful. Another main result is that, for irreducible infinite W of finite rank, the closed imaginary cone is the only non-zero closed pointedW -stable cone contained in the pointed cone spanned by the simple roots (Theorem 7.6). A third main result gives algebraic descriptions of the face lattices of the imaginary cone and Tits cone. Namely, it is shown in Section 11 that (under mild finiteness and non-degeneracy conditions) several notions of faces of the imaginary cone (and of the Tits cone) coincide, and that the face lattice of the imaginary cone is isomorphic to the lattice of facial subgroups of W with no finite components, and is dual to the lattice of faces of the Tits cone. Here, the facial subgroups are defined as the (parabolic) reflection subgroups arising as stabilizers of points of the Tits cone. This result is quite delicate since, in general, neither the imaginary cone nor the Tits cone are closed cones and many of the 2000 Mathematics Subject Classification. 20F55 (Primary) 17B22 (Secondary).