The harmonic complement of the coinvariant algebra

Choose an arbitrary positive definite symmetric bilinear form on ${\mathfrak{h}}_{\mathbb{R}}$ and average it over the finite group $W$, obtaining a $W$-invariant positive definite form $(,)$; choose an orthonormal basis $e_1,\dots ,e_r$ of ${\mathfrak{h}}_{\mathbb{R}}$ for it. On the symmetric algebra $S=\mathbb{C}[e_1,\dots ,e_r]$ let $⟨,⟩$ be the Fischer pairing, the sesquilinear form with $⟨e^{\alpha },e^{\beta }⟩=\alpha !{\delta }_{\alpha \beta }$ on monomials $e^{\alpha }$ ($\alpha$ a multi-index); it is positive definite, and $W$ acts on $S$ by orthogonal transformations for it because the form on ${\mathfrak{h}}_{\mathbb{R}}$ is $W$-invariant. For $p\in S$ let ${\mathrm{\partial }}_p=p({\mathrm{\partial }}_1,\dots ,{\mathrm{\partial }}_r)$ be substitution of the partial derivatives ${\mathrm{\partial }}_i=\mathrm{\partial }\mathrm{/}\mathrm{\partial }{e}_i$; then multiplication by $p$ has adjoint ${\mathrm{\partial }}_{\bar{p}}$, i.e. $⟨pf,g⟩=⟨f,{\mathrm{\partial }}_{\bar{p}}g⟩$, as checked on monomials. Choose real basic invariants: real and imaginary parts of complex invariants are invariant, and one may choose a real homogeneous basis of the indecomposables of the invariant algebra and lift it; its complexification gives basic generators. Complex conjugation preserves $S_{+}^W$, so simultaneous kernels are unchanged. Set ${\mathcal{H}}_d:=I_d^{\perp }$ inside the finite-dimensional space $S_d$, and $\mathcal{H}:={⨁}_d{\mathcal{H}}_d$.

The invariant ring acts on $S$ with the expected Hilbert series: $S$ has a finite homogeneous $S^W$-basis by [F1], say of degrees $m_1,\dots ,m_{\mathrm{\mid }W\mathrm{\mid }}$; their images in $S\mathrm{/}{S}_{+}^{W}S=C$ form a $\mathbb{C}$-basis by graded Nakayama, so ${\sum }_j{t}^{m_j}=\mathrm{Hilb}(C,t)$ and therefore $\mathrm{Hilb}(S,t)=\mathrm{Hilb}(S^W,t)\mathrm{Hilb}(C,t)$.

By step 1.1 the spaces $S_d=I_d\oplus {\mathcal{H}}_d$ are orthogonal direct sums and $\mathcal{H}$ is a graded subspace; it is $W$-stable because $W$ preserves the form and $I$ is a $W$-stable homogeneous ideal, and $\mathcal{H}$ is a graded complement of $I$. Moreover ${\mathcal{H}}_d=\{f\in S_d:{\mathrm{\partial }}_{p}f=0\text{ for all }p\in S_{+}^W\}$: since $I=(w_1,\dots ,w_r)$, one has $f\in I_d^{\perp }$ exactly when $⟨f,w_{i}g⟩=0$ for every $i$ and every $g\in S_{d-2d_i}$, which by adjointness is $⟨{\mathrm{\partial }}_{w_i}f,g⟩=0$ for all such $g$, hence ${\mathrm{\partial }}_{w_i}f=0$ by positive definiteness; and for $p={\sum }_i{q}_i{w}_i$ one has ${\mathrm{\partial }}_p={\sum }_i{\mathrm{\partial }}_{q_i}{\mathrm{\partial }}_{w_i}$, so vanishing for the $w_i$ is vanishing for all of $S_{+}^W$.

The restriction $\rho {\mathrm{\mid }}_{\mathcal{H}}:\mathcal{H}\to C$ is an isomorphism of graded vector spaces because $\rho$ has kernel $I$ and $\mathcal{H}$ is a complement of $I$ by step 1.1; it is a map of $W$-modules because $\rho$ is $W$-equivariant. Hence $\dim {\mathcal{H}}_d=\dim C_d$ for every $d$, and $\dim \mathcal{H}=\dim C=\mathrm{\mid }W\mathrm{\mid }$ by [F2].

Let $\phi :S^W\otimes \mathcal{H}\to S$, $p\otimes h\mapsto ph$. This is a homomorphism of graded $S^W$-modules and of $W$-modules. Its reduction modulo $S_{+}^W$, namely $\mathcal{H}\to C$, $h\mapsto \rho (h)$, is surjective because it is the isomorphism $\rho {\mathrm{\mid }}_{\mathcal{H}}$ of step 2.2.

The map $\phi$ is surjective: given homogeneous $f\in S_d$, choose $h\in {\mathcal{H}}_d$ with $\rho (f)=\rho (h)$ by step 3.1; then $f-h\in I_d={\sum }_i(w_{i}S{)}_d={\sum }_i{w}_i{S}_{d-2d_i}$, so by induction on $d$ each $w_{i}g$ with $g\in S_{d-2d_i}$ lies in $S^W\cdot \mathcal{H}$, and adding $h$ exhibits $f$ in the image.

The map $\phi$ is injective: by steps 1.2 and 2.2 the two sides have the same Hilbert series, $\mathrm{Hilb}(S^W\otimes \mathcal{H},t)=\mathrm{Hilb}(S^W,t)\mathrm{Hilb}(\mathcal{H},t)=\mathrm{Hilb}(S^W,t)\mathrm{Hilb}(C,t)=\mathrm{Hilb}(S,t),$ both are graded with finite-dimensional graded pieces, and by step 4.1 the kernel of the surjective graded $S^W$-linear map $\phi$ would otherwise contribute positive Hilbert series. Hence $\phi$ is an isomorphism.

Steps 2.1, 2.2 and 5.1 give a $W$-stable graded complement $\mathcal{H}$ of $I$ with $\rho {\mathrm{\mid }}_{\mathcal{H}}:\mathcal{H}\cong C$ and $S^W\otimes \mathcal{H}\cong S$; the description of $\mathcal{H}$ as the simultaneous invariant differential kernel follows from the adjointness $⟨pf,g⟩=⟨f,{\mathrm{\partial }}_{\bar{p}}g⟩$ of step 1.1, and $\dim \mathcal{H}=\mathrm{\mid }W\mathrm{\mid }$ by step 2.2. The Axiom of Choice is inherited from the invariant-theory suppliers [F1] and [F2]; the construction itself uses only finite averages and finite-dimensional linear algebra.