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?
4 Answers
If you look at section 4 of Thomas Lam and Pavlo Pylyavskyy's recent preprint, 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.
If you consider just a Weyl group, then I guess there is nothing replacing the longest element (of course, every element also has a reduced decomposition, though).
However, as pointed by Greg Muller, there are some situations when there is a good analogue, namely in the theory of twin buildings. For example, if you consider a semisimple algebraic group G defined over a field K, then G(K[t,t^(1)]) acts on two affine buildings, which are the buildings associated to G(K((t)) ) and G( K((t^(1))) ). Then there is something called "codistance" between chambers of these two buildings. Two chambers at codistance 1 are called "opposite". This opposition relation shares the same kind of property, and should really be thought as an analogue of, the opposition relation in finite buildings and Coxeter complexes.
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 w_{0} 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.


$\begingroup$ apologies to Peter if he is reading this for forgetting his 'n'. $\endgroup$ Oct 21, 2011 at 17:02
I don't know very much about this, but I have heard that in the theory of 'buildings' (nice simplicial complexes on which Coexeter groups act), the affine Weyl groups act naturally on 'twin buildings', a pair of spaces with a notion of distance between them. I'm under the impression a map which switches the two buildings makes a good analog of the longest word.