Canonical left cells in affine Weyl groups (Q1115531)

Let (W,S) be an irreducible affine Weyl group with S the set of Coxeter generators. Define the sets \({\mathcal R}(w)=\{s\in S|\) \(ws<w\}\) and \(Y_ s=\{y\in W|\) \({\mathcal R}(y)\subset \{s\}\}\) for any \(w\in W\) and \(s\in S\). An element \(s\in S\) is said to be special if the subgroup of W generated by S-\(\{\) \(s\}\) has maximum possible order. Then in the present paper, the authors show that for any \(s\in S\), the intersection of the set \(Y_ s\) with any two-sided cell \(\Omega\) of W is nonempty and is a union of some left cells of W. This result becomes more accurate in two cases. One is that when s is special, \(Y_ s\cap \Omega\) is exactly one left cell of W. The other is that when \(\Omega\) is the lowest two-sided cell of W, \(Y_ s\cap \Omega\) consists of n(s) left cells of W, where \(n(s)=\)order of subgroup generated by \(S-\{s_ 0\}/\)order of subgroup generated by \(S-\{s\}\) and \(s_ 0\) is a special element of S.