Center and Representations of Infinitesimal Hecke Algebras of 𝔰𝔩₂

Abstract

In this article, we compute the center of the infinitesimal Hecke algebras H _z associated to 𝔰𝔩₂; 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 Duflo's theorem for H _z .

Key Words:

• Center
• Commutator
• Maximal vector
• Primitive ideal

2000 Mathematics Subject Classification:

• Primary 16D90
• Secondary 16S80, 17B20

ACKNOWLEDGMENTS

We thank the referee for his patient reading, and for making numerous suggestions and comments, which helped in improving this manuscript.

This article is essentially an improved write-up of A.T.'s Ph.D. thesis from the University of Chicago, and Section 2 is a more detailed exposition of an earlier preprint [10].

Notes

Communicated by K. Misra.