The coinvariant algebra is the graded regular representation of W

By [F1], $\rho {\mathrm{\mid }}_{\mathcal{H}}:\mathcal{H}\stackrel{\sim }{\to }C$ is an isomorphism of graded $W$-modules, so the graded traces agree: $\mathrm{HTr}(C,t)(w)=\mathrm{HTr}(\mathcal{H},t)(w)$ for every $w\in W$.

Molien's formula for the graded trace of $S$: for $\mathrm{\mid }t\mathrm{\mid }<1$ and $w\in W$ with eigenvalues ${\lambda }_1,\dots ,{\lambda }_r$ on ${\mathfrak{h}}^{\ast }$, one has $\mathrm{tr}(w\mid S_i)=h_i({\lambda }_1,\dots ,{\lambda }_r)$, the complete homogeneous symmetric polynomial of degree $i$, whose generating function is ${\sum }_{i\ge 0}{h}_i(\lambda )t^i={\prod }_j(1-t{\lambda }_j{)}^{-1}$; with page degrees doubled this gives $\mathrm{HTr}(S,t)(w)=\det (1-t^{2}w\mid {\mathfrak{h}}^{\ast }{)}^{-1}$ as an identity of formal power series, the finite sum over the eigenvalues making every coefficient finite.

Since $S\cong S^W\otimes \mathcal{H}$ with $W$ acting trivially on the polynomial ring $S^W=\mathbb{C}[w_1,\dots ,w_r]$, whose homogeneous piece of page degree $2D$ has dimension equal to the number of monomials of weighted degree $D$, the graded trace factors as $\mathrm{HTr}(S,t)(w)={\prod }_i(1-t^{2d_i}{)}^{-1}\cdot \mathrm{HTr}(\mathcal{H},t)(w)$ for $\mathrm{\mid }t\mathrm{\mid }<1$.

Combining steps 1.1 to 1.3, $\mathrm{HTr}(C,t)(w)={\prod }_{i=1}^r(1-t^{2d_i})\cdot \mathrm{HTr}(S,t)(w)={\prod }_{i=1}^r(1-t^{2d_i})\cdot \det (1-t^{2}w\mid {\mathfrak{h}}^{\ast }{)}^{-1},$ first as an identity of formal power series for $\mathrm{\mid }t\mathrm{\mid }<1$ and then as an identity of rational functions; this is assertion (2).

Limits at $t\to 1$: each factor $1-t^{2d_i}$ vanishes to order exactly one at $t=1$ because its derivative there is $-2d_i\mathrm{\ne }0$, so the numerator vanishes to order $r$. The denominator $\det (1-t^{2}w\mid {\mathfrak{h}}^{\ast })={\prod }_j(1-t^2{\lambda }_j)$ vanishes at $t=1$ to the order of the multiplicity of the eigenvalue $1$ of $w$ on ${\mathfrak{h}}^{\ast }$, that is, to order $\dim {\mathfrak{h}}^w$; the reflection representation is faithful, so $w\mathrm{\ne }1$ implies ${\mathfrak{h}}^w\mathrm{\ne }\mathfrak{h}$ and $\dim {\mathfrak{h}}^w\le r-1$, whence for $w\mathrm{\ne }1$ the limit of $\mathrm{HTr}(C,t)(w)$ is $0$. At $w=1$ the quotient is ${\prod }_i(1-t^{2d_i})\mathrm{/}(1-t^2{)}^r\to {\prod }_i{d}_i=\mathrm{\mid }W\mathrm{\mid }$ by [F2].

The graded trace at $t=1$ is the ordinary character of the finite-dimensional $W$-module $C$, so step 3.1 shows ${\chi }_C(w)=\mathrm{\mid }W\mathrm{\mid }{\delta }_{w,1}$, the character of the regular representation; by the orthonormality of irreducible characters [F3], the multiplicity of each irreducible $E$ in $C$ equals its multiplicity in $\mathbb{C}W$, namely $\dim E$. Hence $C\cong \mathbb{C}W$ as $W$-modules, which is assertion (1).

Graded multiplicities: each homogeneous piece of $C$ decomposes into isotypic components, giving each irreducible $E$ its uniquely determined graded multiplicity series ${\mathrm{mult}}_E(C,t)$, and the identities ${\sum }_E{\mathrm{mult}}_E(C,t){\chi }_E(w)=\mathrm{HTr}(C,t)(w)$ for all $w$ determine ${\mathrm{mult}}_E(C,t)$ as the coefficient of ${\chi }_E$ in the expansion of the graded character in the orthonormal basis of [F3]; setting $t=1$ recovers ${\mathrm{mult}}_E(C,1)$ as the multiplicity of $E$ in $C$, which is $\dim E$ by step 4.1; this is assertion (3). The Hilbert-series specialization of step 2.1 at $w=1$ is ${\prod }_i(1-t^{2d_i})\mathrm{/}(1-t^2{)}^r={\prod }_i(1+t^2+\cdots +t^{2(d_i-1)})$.