Reflectable bases for affine reflection systems

Abstract

The notion of a “root base” together with its geometry plays a crucial role in the theory of finite and affine Lie theory. However, it is known that such a notion does not exist for the recent generalizations of finite and affine root systems such as extended affine root systems and affine reflection systems. In this work, we consider the notion of a “reflectable base” for an affine reflection system R. A reflectable base for R is a minimal subset Π of roots such that the non-isotropic part of the root system can be recovered by reflecting roots of Π relative to the hyperplanes determined by Π. We give a full characterization of reflectable bases for tame irreducible affine reflection systems of reduced types, excluding types E6,7,8. As a by-product of our results, we show that if the root system under consideration is locally finite, then any reflectable base is an integral base.