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Lusztig’s \(a\)-function in type \(B_n\) in the asymptotic case. (English) Zbl 1173.20301
Summary: We study Lusztig’s \(\mathbf a\)-function for a Coxeter group with unequal parameters. We determine that function explicitly in the “asymptotic case” in type \(B_n\), where the left cells have been determined in terms of a generalized Robinson-Schensted correspondence by BonnafĂ© and the second author. As a consequence, we can also show that all of Lusztig’s conjectural properties (P1)-(P15) hold in this case, except possibly (P9), (P10) and (P15). Our methods rely on the “leading matrix coefficients” introduced by the first author. We also interprete the ideal structure defined by the two-sided cells in the associated Iwahori-Hecke algebra \(\mathcal H_n\) in terms of the Dipper-James-Murphy basis of \(\mathcal H_n\).

MSC:
20C08 Hecke algebras and their representations
20F55 Reflection and Coxeter groups (group-theoretic aspects)
05E15 Combinatorial aspects of groups and algebras (MSC2010)
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[1] Lecture Notes in Math. 1024 pp 99– (1983)
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