Split Spetses for primitive reflection groups

Details Split Spetses for primitive reflection groups Split Spetses for primitive reflection groups

2012-04-26

arxiv.org/abs/1204.5846v1

Mathematics

Representation Theory

146 pages

Scientific paper

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).

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