Let $V$ be a complex vector space of finite dimension $n$ and let $W$ be a finite unitary reflection group. This is, $W$ is a subgroup of $GL(V)$ generated by reflections, i.e., elements $r \in GL(V)$ for which $\operatorname{dim}(\operatorname{ker}(r-1)) = n-1$.

It is well-known that a finite subgroup of $GL(V)$ is a unitary reflection group if and only if $\mathbb{C}[V]^W \cong \mathbb{C}[f_1,\ldots,f_n]$ is again a polynomial algebra generated by homogeneous polynomials $f_1,\ldots,f_n$.

In this case, it is also well known that the coinvariant algebra $\mathbb{C}[V]^{\operatorname{coW}} = \mathbb{C}[V] \big/ \langle f_1,\ldots,f_n\rangle$ carries the regular representation as a $W$-module. This is, every $d$-dimensional irreducible representation of $W$ is found $d$ times inside $\mathbb{C}[V]^{\operatorname{coW}}$.

All this can be found e.g. in the Lehrer-Taylor book "Unitary reflection groups" in Chapter 3.

In Proposition 3.23, the result about the regular representation is obtained by showing that $\mathbb{C}[V]^{\operatorname{coW}}$ has dimension $|W|$ and the character vanishes for elements $\neq 1 \in W$.

My question now is

Is it known how to actually construct the $W$-equivariant isomorphisms between an irreducible representation $\Lambda$ of $W$ and its $\operatorname{dim}(\Lambda)$ many copies inside $\mathbb{C}[V]^{\operatorname{coW}}$ ?

A pointer to any treatment (being it only the symmetric group, real reflection groups, or more general situations) would be much appreciated!