Ordering families using Lusztig’s symbols in type $$B$$ B : the integer case

Abstract

Let Irr(W ) be the set of irreducible representations of a finite Weyl group W . Following an idea from Spaltenstein, Geck has recently introduced a preorder ≤ L on Irr(W ) in connection with the notion of Lusztig families. In a later paper with Iancu, they have shown that in type B (in the asymptotic case and in the equal parameter case) this order coincides with the order on Lusztig symbols as defined by Geck and the second author in [4]. In this paper, we show that this caracterisation extends to the so-called integer case, that is when the ratio of the parameters is an integer.