Dirac induction for rational Cherednik algebras

TL;DR

This paper develops the theory of Dirac operators for rational Cherednik algebras, establishing their indices, relations to module characters, and introducing integral-reflection modules with parameter-independent indices, advancing representation theory.

Contribution

It introduces local and global Dirac indices for rational Cherednik algebras, linking them to module characters and defining new integral-reflection modules with parameter-independent indices.

Findings

Established relations between Dirac index and module characters.

Defined and computed Dirac operator indices on integral-reflection modules.

Introduced dualised generalised Dunkl-Opdam operators.

Abstract

We introduce the local and global indices of Dirac operators for the rational Cherednik algebra $\mathsf{H}_{t,c}(G,\mathfrak{h})$, where $G$ is a complex reflection group acting on a finite-dimensional vector space $\mathfrak{h}$. We investigate precise relations between the (local) Dirac index of a simple module in the category $\mathcal{O}$ of $\mathsf{H}_{t,c}(G,\mathfrak{h})$, the graded $G$-character of the module, the Euler-Poincar\'e pairing, and the composition series polynomials for standard modules. In the global theory, we introduce integral-reflection modules for $\mathsf{H}_{t,c}(G,\mathfrak{h})$ constructed from finite-dimensional $G$-modules. We define and compute the index of a Dirac operator on the integral-reflection module and show that the index is, in a sense, independent of the parameter function $c$. The study of the kernel of these global Dirac operators leads naturally to a notion of dualised generalised Dunkl-Opdam operators.