Imaginary cone and reflection subgroups of Coxeter groups

TL;DR

This paper investigates the structure of the imaginary cone in Coxeter groups, revealing its properties, faces, and duality relations, with implications for algebraic and combinatorial aspects of these groups.

Contribution

It provides a detailed analysis of the imaginary cone for Coxeter groups, including its containment relations, face structure, and duality, extending understanding of these geometric objects.

Findings

Imaginary cone of reflection subgroups is contained in that of the whole group.

For irreducible infinite Coxeter groups, the closed imaginary cone is unique and pointed.

The face lattice of the imaginary cone is explicitly described and related to reflection subgroups.

Abstract

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.