On Harish-Chandra's Isomorphism

TL;DR

This paper reviews Harish-Chandra's isomorphism, its applications, and introduces nonsymmetric shift operators for arbitrary root systems with a transmutation property related to Dunkl-Cherednik operators.

Contribution

It announces the existence and uniqueness of nonsymmetric shift operators for any root system, extending hypergeometric shift operators.

Findings

Nonsymmetric shift operators exist and are unique for all root systems.

These operators have a transmutation property with Dunkl-Cherednik operators.

On symmetric functions, they reduce to known hypergeometric shift operators.

Abstract

This is the text of a talk given by the first author at the Harish-Chandra centenary meeting held in Allahabad in October 2023. It reviews Harish-Chandra's isomorphism and its many applications to representation theory and mathematical physics. It also announces the existence and uniqueness of nonsymmetric shift operators for an arbitrary root system. These are differential-reflection operators with a transmutation property relative to Dunkl-Cherednik operators: they shift the parameter k of these operators by 1, and restrict on symmetric functions to the hypergeometric shift operators introduced by the first author.