On the determination of Kazhdan-Lusztig cells in affine Weyl groups with unequal parameters

Details On the determination of Kazhdan-Lusztig cells in affine Weyl groups with unequal parameters On the determination of Kazhdan-Lusztig cells in affine Weyl groups with unequal parameters

2007-02-09

arxiv.org/abs/math/0702272v5

Mathematics

Representation Theory

22 pages, 2 figures, the revised version contains additional reference and some rewritting. Submitted

Scientific paper

Let W be a Coxeter group and L be a weight function on W. Following Lusztig, we have a corresponding decomposition of W into left cells, which have important applications in representation theory. We study the case where $W$ is an affine Weyl group of type $\tilde{G_{2}}$. Using explicit computation with \textsf{CHEVIE}, we show that (1) there are only finitely many possible decompositions into left cells and (2) the number of left cells is finite in each case, thus confirming some of Lusztig's conjectures in this case. For the proof, we show some equalities on the Kazhdan-Lusztig polynomials which hold for any affine Weyl groups.

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