Dirac Induction for Rational Cherednik Algebras

We introduce the local and global indices of Dirac operators for the rational Cherednik algebra H-t,H-c(G, h), where G is a complex reflection group acting on a finite-dimensional vector space h. We investigate precise relations between the (local) Dirac index of a simple module in the category O of H-t,H-c(G, h), the graded G-character of the module, the Euler-Poincare pairing, and the composition series polynomials for standard modules. In the global theory, we introduce integral-reflection modules for H-t,H-c(G, 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.