  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.
