  We define Lie subalgebras of the group algebra of a finite pseudo-reflection
group that are involved in the definition of the Cherednik KZ-systems, and
determine their structure. We provide applications for computing the Zariski
closure of the image of generalized (pure) braid group B inside the
representations of the corresponding Hecke algebras. We also get
unitarizability results for the representations of B originating from Hecke
algebras for suitable parameters, and relate our Lie algebras with the
topological closure of B in these compact cases.
