The Dunkl Weight Function for Rational Cherednik Algebras

In this paper we prove the existence of the Dunkl weight function Kc,λ for any irreducible representation λ of any finite Coxeter group W , generalizing previous results of Dunkl. In particular, Kc,λ is a family of tempered distributions on the real reflection representation of W taking values in EndC(λ), with holomorphic dependence on the complex multi-parameter c. When the parameter c is real, the distribution Kc,λ provides an integral formula for Cherednik’s Gaussian inner product γc,λ on the Verma module ∆c(λ) for the rational Cherednik algebra Hc(W, h). In this case, the restriction of Kc,λ to the hyperplane arrangement complement hR,reg is given by integration against an analytic function whose values can be interpreted as braid group invariant Hermitian forms on KZ(∆c(λ)), where KZ denotes the Knizhnik-Zamolodchikov functor introduced by Ginzburg-Guay-OpdamRouquier. This provides a concrete connection between invariant Hermitian forms on representations of rational Cherednik algebras and invariant Hermitian forms on representations of Iwahori-Hecke algebras, and we exploit this connection to show that the KZ functor preserves signatures, and in particular unitarizability, in an appropriate sense.