Infinitesimal Hecke Algebras III

TL;DR

This paper introduces Lie subalgebras associated with finite pseudo-reflection groups, explores their structure, and applies these findings to analyze braid group representations, their Zariski closures, and unitarizability properties.

Contribution

It defines new Lie subalgebras linked to Cherednik KZ-systems and investigates their structure and applications in braid group representation theory.

Findings

Determined the structure of Lie subalgebras related to pseudo-reflection groups.

Computed the Zariski closure of braid group images in Hecke algebra representations.

Established unitarizability results for certain braid group representations.

Abstract

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.