Two families of Dirac-like operators for Drinfeld's Hecke algebra

TL;DR

This paper introduces two new types of Dirac-like operators for Drinfeld's Hecke algebra, exploring their properties and potential to determine infinitesimal characters, with explicit examples for type A.

Contribution

It defines Parthasarathy and warped Dirac operators for Drinfeld's Hecke algebra, extending Dirac theory and analyzing their cohomological properties.

Findings

Parthasarathy operators satisfy a Dirac inequality.

Warped Dirac operators ensure non-zero cohomology for unitary modules.

Explicit families of operators are constructed for type A Hecke algebra.

Abstract

In this paper, we define two generalisations of Dirac operators for Drinfeld's Hecke algebra. One generalisation, Parthasarathy operators inherit the notion of the Dirac inequality. The second generalisation, warped Dirac operators are such that every unitary module must have a non-zero warped Dirac cohomology. An open question is whether non-zero warped Dirac cohomology can determine the infinitesimal character akin to the fact that non-zero Dirac cohomology does. For a type $A$ Hecke algebra we give a family of operators in each class.