REFLECTION GROUPS, REFLECTION ARRANGEMENTS, AND INVARIANT REAL VARIETIES

摘要

Let X be a nonempty real variety that is invariant under the action of a reflection group G. We conjecture that if X is defined in terms of the first k basic invariants of G (ordered by degree), then X meets a k-dimensional flat of the associated reflection arrangement. We prove this conjecture for the infinite types, reflection groups of rank at most 3, and F-4 and we give computational evidence for H-4. This is a generalization of Timofte's degree principle to reflection groups. For general reflection groups, we compute nontrivial upper bounds on the minimal dimension of flats of the reflection arrangement meeting X from the combinatorics of parabolic subgroups. We also give generalizations to real varieties invariant under Lie groups.