Dirac cohomology and simple modules of the Dunkl-Opdam subalgebra via inherited Drinfeld properties

TL;DR

This paper introduces a new presentation of the Dunkl-Opdam subalgebra, proving it is a Drinfeld algebra, and develops Dirac cohomology and a Langlands classification for it, advancing understanding of its module structure.

Contribution

It provides a novel presentation of the Dunkl-Opdam subalgebra as a Drinfeld algebra and establishes Dirac cohomology and Langlands classification for it.

Findings

Dunkl-Opdam subalgebra shown to be a Drinfeld algebra

Defined Dirac cohomology for the subalgebra

Formalized generalized graded Hecke algebras and their classification

Abstract

In this paper we define a new presentation for the Dunkl-Opdam subalgebra of the rational Cherednik algebra. This shows that the Dunkl-Opdam subalgebra is a Drinfeld algebra. We use this fact to define Dirac cohomology for the DO subalgebra. We also formalise generalised graded Hecke algebras and define a Langlands classification to generalised graded Hecke algebras.