Partial orderings of an affine Weyl group

In this note we draw together some of the scattered literature dealing with several partial orderings of affine Weyl groups. Most of the theory was developed as a tool in the study of modular representations for groups of Lie type, but here we focus just on an affine Weyl group Wa in its elementary geometric setting while sometimes invoking also its structure as a Coxeter group. While notation and terminology vary considerably in the cited sources, the conventions here usually follow [6]. I’m grateful to Florian Herzig for calling my attention to problematic references needed for applications of modular representation theory to his current joint preprint posted at [5]. The help of a colleague in translating Jian-pan Wang’s paper [13] into English is also appreciated.