This is a continuation of similar small groups research at CIRM from 2012 to 2015, which produced a report « Split Spetses for primitive reflection groups » which appeared as an Asterisque volume in 2014.
Lusztig’s work has shown that quite a few properties of finite reductive groups can be combinatorially computed from the Weyl group, or in the case of a twisted group from the corresponding reflection coset : for instance, the unipotent degrees, or the Fourier matrix.
The research line started by Broué, Malle and Michel in 1993 during a conference in Spetses island (Greece) is to try the same constructions replacing the Weyl group by a complex reflection group. This works for the class of groups called Spetsial.
We plan to focus on the Fourier matrices and their categorification. In the case of reductive group, each Lusztig family of unipotent characters is attached to a small group, such that the category of representations of its Drinfeld double parameterizes the representations in the family.
Any modular category gives similarly a matrix. Some of the matrices appearing in Spetses have been explained by such « exotic » categories. Recently Bonnafe and Rouquier have explained matrices appearing in the cyclic group from a stable representation category of a quantum group.
Our goal is to complete and organize the current state of our knowledge in order to write another comprehensive report.