  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.
