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Constructible characters and \(b\)-invariant. (English) Zbl 1328.20008
Summary: If \(W\) is a finite Coxeter group and \(\varphi\) is a weight function, Lusztig has defined \(\varphi\)-constructible characters of \(W\), as well as a partition of the set of irreducible characters of \(W\) into Lusztig \(\varphi\)-families. We prove that every Lusztig \(\varphi\)-family contains a unique character with minimal \(b\)-invariant, and that every \(\varphi\)-constructible character has a unique irreducible constituent with minimal \(b\)-invariant. This generalizes Lusztig’s result about special characters to the case where \(\varphi\) is not constant. This is compatible with some conjectures of Rouquier and the author about Calogero-Moser families and Calogero-Moser cellular characters.

MSC:
20C08 Hecke algebras and their representations
20F55 Reflection and Coxeter groups (group-theoretic aspects)
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Full Text: Euclid