A counter-example to a conjecture of Lusztig.
In [Adv. Stud. Pure Math. 6, 255-287 (1985; Zbl 0569.20032)], \textit{G. Lusztig} defined a function \(a\) on a Coxeter group \(W\). This function plays an important role in the study of the Hecke algebra of \(W\); in particular, Lusztig proved that for \(W\) affine or finite, the function \(a\) is constant on the two-sided cells of \(W\). Later on, in the book [\textit{G. Lusztig}, Hecke algebras with unequal parameters. CRM Monograph Series 18. Providence: AMS (2003; Zbl 1051.20003)], he defined another function \(a'\) on \(W\), related to the function \(a\), and conjectured that these two functions \(a\) and \(a'\) are in fact equal. Together with some other conjecture, this would imply that there are only finitely many two-sided cells in \(W\). This paper disproves this conjecture of Lusztig, by exhibiting a counter-example. The group \(W\) in this example is the affine Coxeter group \(\widetilde A_{10}\), which is seen as a group of permutations of \(\mathbb{Z}\), as in [\textit{J.-y. Shi}, The Kazhdan-Lusztig cells in certain affine Weyl groups. Lecture Notes in Mathematics 1179. Berlin etc.: Springer-Verlag (1986; Zbl 0582.20030)]. The author describes explicitly an element \(w\in W\) for which \(a(w)\neq a'(w)\). Finally, the author suggests another conjecture about the structure on two-sided cells of Coxeter groups. Assuming some other conjectures of Lusztig, this new conjecture is weaker than the previous one. Then he checks this new conjecture on Coxeter groups of type \(\widetilde A_n\) and \(\widetilde D_4\).
scientific article; zbMATH DE number 5716403
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