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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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