TITLE: Longest Element of an Affine Weyl Group
QUESTION [9 upvotes]: I know that the Weyl groups of affine Lie algebras don't have a longest element, but are there any good substitutes for w_0. In particular, is there any good substitute for a reduced decomposition of the longest element?

REPLY [5 votes]: Here is another notion of a reduced decomposition for the long word, as used for example in a recent preprint of Baumann, Kamnitzer and Tingley.
Recall the theorem from the finite type case that reduced decompositions of w0 are in bijection with convex orderings on the set of positive roots. The latter notion is easier to generalise, and can be considered an appropriate analogue.
Definition: A convex order on the set of all positive roots Φ+ is a preorder ≤ such that
i) For all α, β in Φ+, we have α ≤ β or β ≤ α (OR not XOR).
ii) If α ≤ β and α+β is a root, then α ≤ β+α ≤ β.
iii) If α ≤ β and β ≤ α then α and β are proportional.
Voila.