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?



Longest Element of an Affine Weyl Group
If you look at section 4 of Thomas Lam and Pavlo Pylyavskyy's [recent preprint](https://arxiv.org/abs/0906.0610), they study infinite reduced words in W, modulo braid and commutation relations. This is a good substitute for a reduced word for the long word in the context of total positivity, as they explain. I think it should also be useful in the context of canonical bases.


