Cyclotomic Root Systems

Abstract: 

I will speak about joint work with Michel Broué and Jean Michel (https://arxiv.org/abs/1704.03779) which is motivated by questions coming from theSpetses project. For GG  a (complex) reflection group defined on a vector space VV over a number field kk, we define a ZkZk-root system for GG, where ZkZk denotes the ring of integers of kk. The definition generalises the notion of a ZZ-root system for a Weyl group; roots are no longer vectors, but rank one ZkZk-submodules of VV. The definition has natural consequences, such as that restricting to a parabolic subgroup gives rise to a root system for the parabolic. In this way, for example, Z[i]Z[i]-root systems naturally arise for Weyl groups of type BB, different from the Weyl types BB and CC. We classify root systems, as well as root and coroot lattices, for complex reflection groups over the field of definition and present corresponding Cartan matrices. If time permits, I will describe how in the case of spetsial groups, what we define to be the connection index has a property which generalises the situation in Weyl groups.