Constructible characters and ${\boldsymbol{b}}$-invariant

Abstract

If $W$ is a finite Coxeter group and $\varphi$ is a weight function, Lusztig has defined {\it $\\varphi$-constructible characters} of $W$, as well as a partition of the set of irreducible characters of $W$ into {\it 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 {\it special characters} to the case where $\varphi$ is not constant. This is compatible with some conjectures of Rouquier and the author about {\it Calogero-Moser families} and {\it Calogero-Moser cellular characters}.