Familles de caracteres de groupes de reflexions complexes
Gunter Malle, Raphael Rouquier
TL;DR
This work extends Lusztig's partition of irreducible characters from Weyl groups to complex reflection groups by identifying Lusztig families with blocks of the Iwahori–Hecke algebra over a suitable ring ${\mathcal{O}}$, and proves a compatibility between these families and Harish-Chandra $d$-series for unipotent characters. It provides a case-by-case determination of ${\mathcal{O}}$-blocks for spetsial exceptional groups (e.g., $G_4$–$G_{37}$), including partial decomposition matrices for bad primes, using algorithmic induction over parabolic subgroups and Chevie-based computations. A central result shows that inside a $d$-series, the Hecke-family blocks are the intersections of Lusztig families with the $d$-series, generalizing Ro’s $d=1$ principal-series compatibility to broader settings. The paper also furnishes explicit data and classifications of decomposition types (e.g., ${\alpha}_{1^2}(3)$, ${\mathcal{M}}(Z_3)$) and demonstrates the practical feasibility of these computations in the spetsial regime, linking Hecke-theoretic block structure to unipotent representations of finite reductive groups.
Abstract
Nous etudions certains types de blocs d'algebres de Hecke associees aux groupes de reflexions complexes qui generalisent les familles de caracteres definies par Lusztig pour les groupes de Weyl. Nous determinons ces blocs pour les groupes de reflexions spetsiaux et nous etablissons un theoreme de compatibilite entre familles et d-series de Harish-Chandra. ----- We study certain types of blocks of Hecke Algebras associated to complex reflection groups which generalize the families of characters defined by Lusztig for Weyl groups. We determine these blocks for the special reflection groups and we establish a compatibility theorem between families and d-series of Harish-Chandra.