HAL : hal-00691314, version 1

arXiv : 1204.5846

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Split Spetses for primitive reflection groups

Michel Broué 1, Gunter Malle 2, Jean Michel 1

(24/04/2012)

Let $(V,W)$ be an exceptional spetsial irreducible reflection group $W$ on a complex vector space $V$, that is a group $G_n$ for $n \in \{ 4, 6, 8, 14, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37\}$ in the Shephard-Todd notation. We describe how to determine some data associated to the corresponding (split) ''spets'', given complete knowledge of the same data for all proper subspetses (the method is thus inductive). The data determined here is the set Uch$(\mathbb G)$ of ''unipotent characters'' of $\mathbb G$ and the associated set of Frobenius eigenvalues, and its repartition into families. The determination of the Fourier matrices linking unipotent characters and ''unipotent character sheaves'' will be given in another paper. The approach works for all split reflection cosets for primitive irreducible reflection groups. The result is that all the above data exist and are unique (note that the cuspidal unipotent degrees are only determined up to sign).

1 :	Institut de Mathématiques de Jussieu (IMJ)
	CNRS : UMR7586 – Université Pierre et Marie Curie [UPMC] - Paris VI – Université Paris VII - Paris Diderot
2 :	FB Mathematik, TU Kaiserslautern
	TU Kaiserslautern

Équipe de recherche : IUF et Université Paris 7

Domaine

:

Mathématiques/Théorie des représentations

Mots Clés : Spetses – reflection cosets – complex reflection groups – unipotent characters.

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Contributeur : Jean Michel <>

Soumis le : Mercredi 25 Avril 2012, 21:47:29

Dernière modification le : Jeudi 26 Avril 2012, 09:11:05