Split Spetses for Primitive Reflection Groups

Résumé du livre

Let $W$ be an exceptional spetsial irreducible reflection group acting on a complex vector space $V$, i.e., 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. The authors describe how to determine some data associated to the corresponding (split) ``spets'' $\mathbb{G} =(V,W)$, given complete knowledge of the same data for all proper subspetses (the method is thus inductive). The data determined here are the set $\mathrm{Uch}(\mathbb{G})$ of ``unipotent characters'' of $\mathbb{G}$ and its repartition into families, as well as the associated set of Frobenius eigenvalues. 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). The authors keep track of the complete list of axioms used. In order to do that, they explain in detail some general axioms of ``spetses'', generalizing (and sometimes correcting) along the way.