Split Spetses for primitive reflection groups

TL;DR

This paper develops an inductive method to determine unipotent characters and Frobenius eigenvalues for split spetses associated with exceptional primitive reflection groups, establishing their existence and uniqueness.

Contribution

It introduces a new inductive approach to compute unipotent data for split spetses of primitive reflection groups, extending the understanding of their representation theory.

Findings

Data for unipotent characters and Frobenius eigenvalues are determined and shown to be unique.

The method applies to all split reflection cosets of primitive irreducible reflection groups.

Existence and uniqueness of the unipotent data are established.

Abstract

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).