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.

52 pages