Constructible characters and \(b\)-invariant. (English) Zbl 1328.20008

Bonnafé, Cédric

Bull. Belg. Math. Soc. - Simon Stevin 22, No. 3, 377-390 (2015).

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)

Keywords:

finite Coxeter groups; \(b\)-invariants; constructible characters; cellular characters; Lusztig families; irreducible constituents