The Dunkl Weight Function for Rational Cherednik Algebras

Abstract

In this paper we prove the existence of the Dunkl weight function $K_{c, \lambda}$ for any irreducible representation $\lambda$ of any finite Coxeter group $W$, generalizing previous results of Dunkl. In particular, $K_{c, \lambda}$ is a family of tempered distributions on the real reflection representation of $W$ taking values in $\text{End}_\mathbb{C}(\lambda)$, with holomorphic dependence on the complex multi-parameter $c$. When the parameter $c$ is real, the distribution $K_{c, \lambda}$ provides an integral formula for Cherednik's Gaussian inner product $\gamma_{c, \lambda}$ on the Verma module $\Delta_c(\lambda)$ for the rational Cherednik algebra $H_c(W, \mathfrak{h})$. In this case, the restriction of $K_{c, \lambda}$ to the hyperplane arrangement complement $\mathfrak{h}_{\mathbb{R}, reg}$ is given by integration against an analytic function whose values can be interpreted as braid group invariant Hermitian forms on $KZ(\Delta_c(\lambda))$, where $KZ$ denotes the Knizhnik-Zamolodchikov functor introduced by Ginzburg-Guay-Opdam-Rouquier. 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.