Infinitesimal Hecke Algebras II

TL;DR

This paper investigates the algebraic and topological properties of braid group images within Iwahori-Hecke algebras associated with finite reflection groups, revealing their reductive Lie algebra structure and unitarizability conditions.

Contribution

It characterizes the Zariski closure of braid group images in Iwahori-Hecke algebras and analyzes their Lie algebra structure and unitarizability for Coxeter groups.

Findings

The Zariski closure's Lie algebra is reductive and generated by reflections.

Decomposition of the Lie algebra into simple factors is determined.

Representations are unitarizable when parameters are near 1 with modulus 1.

Abstract

For W a finite (2-)reflection group and B its (generalized) braid group, we determine the Zariski closure of the image of B inside the corresponding Iwahori-Hecke algebra. The Lie algebra of this closure is reductive and generated in the group algebra of W by the reflections of W. We determine its decomposition in simple factors. In case W is a Coxeter group, we prove that the representations involved are unitarizable when the parameters of the representations have modulus 1 and are close to 1. We consequently determine the topological closure in this case.