Degree and valuation of the Schur elements of cyclotomic Hecke algebras

TL;DR

This paper demonstrates that the degree and valuation of Schur elements in cyclotomic Hecke algebras are constant across families of complex reflection groups, extending previous results to exceptional groups.

Contribution

It generalizes the constancy of Schur element invariants to all exceptional complex reflection groups, building on Lusztig and Rouquier's frameworks.

Findings

Degree and valuation are constant on families for exceptional groups

Extends known results from infinite series to all exceptional groups

Supports the broader theory of families in complex reflection groups

Abstract

Following the generalization of the notion of families of characters, defined by Lusztig for Weyl groups, to the case of complex reflection groups, thanks to the definition given by Rouquier, we show that the degree and the valuation of the Schur elements (functions A and a) remain constant on the "families" of the cyclotomic Hecke algebras of the exceptional complex reflection groups. The same result has already been obtained for the groups of the infinite series and for some special cases of exceptional groups.