Center and representations of infinitesimal Hecke algebras of sl_2

TL;DR

This paper computes the center of infinitesimal Hecke algebras related to sl_2, explores their representations in category O, and extends results to gl(n) and sp(2n), including a Duflo-type theorem.

Contribution

It provides the first detailed analysis of the center and representation theory of infinitesimal Hecke algebras for sl_2, gl(n), and sp(2n), and proves a Duflo analogue.

Findings

Computed the center of Hz for sl_2

Studied representations in category O using the center

Proved a Duflo-type theorem for Hz

Abstract

In this paper, we compute the center of the infinitesimal Hecke algebras Hz associated to sl_2 ; then using nontriviality of the center, we study representations of these algebras in the framework of the BGG category O. We also discuss central elements in infinitesimal Hecke algebras over gl(n) and sp(2n) for all n. We end by proving an analogue of the theorem of Duflo for Hz.