Reflection groups, arrangements, and invariant real varieties

Abstract

Let <i>X</i> be a nonempty real variety that is invariant under the action of a reflection group <i>G</i>. We conjecture that if <i>X</i> is defined in terms of the first <i>k</i> basic invariants of <i>G</i> (ordered by degree), then <i>X</i> meets a <i>k</i>-dimensional flat of the associated reflection arrangement. We prove this conjecture for the infinite types, reflection groups of rank at most <i>3</i>, and <i>F₄</i> and we give computational evidence for <i>H₄</i>. 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 <i>X</i> from the combinatorics of parabolic subgroups. We also give generalizations to real varieties invariant under Lie groups.