invariant space, complex reflexion group

Question

I am reading an article written by Pavel Etingof: "Symplectic reflection algebra, Calogero-Moser-Space and deformed Harish Chandra homomorphism". I am trying to figure out the isomorphism (4.15, page 39): $\mathbb{C}[V \oplus V^{*}] \simeq \mathbb{C}[W \times W] \otimes (\mathbb{C}[V]^{W} \otimes \mathbb{C}[V^{*}]^{W})$ where $V$ is $\mathbb{C}$- vector space , $V^{*}$ the dual vector space, $W$ a finite complex reflection group. I don't how to prove this.

Answer 1

Given a finite-dimensional $\mathbf{C}$-vector space $V$, the ring $\mathbf{C}[V^*] \cong \mathrm{Sym}(V)$ of polynomial functions on the dual space may be identified with the ring of constant-coefficient differential operators on $V$ via identifying a vector $v \in V$ with the operator $\partial_v$ of partial differentiation in the direction $v$.

For a finite group $W \subseteq \mathrm{GL}(V)$ of linear transformations of a finite-dimensional $\mathbf{C}$-vector space $V$, define the space of W-harmonic polynomials by $$H_W=\{f \in \mathbf{C}[V] \ | \ d(f)=0 \quad \hbox{for all $d \in \mathbf{C}[V^*]^W_{>0}$}\},$$ where $\mathbf{C}[V]^W_{>0}$ denotes the space of $W$-invariant differential operators with constant term equal to zero.

There is a natural map $$H_W \otimes \mathbf{C}[V]^W \to \mathbf{C}[V], \quad f \otimes g \mapsto fg,$$ that is an isomorphism if and only if $W$ is a complex reflection group. Moreover, in this case as a $\mathbf{C} W$-module, $H_W$ is isomorphic to the regular representation $\mathbf{C} W \cong H_W$ (see e.g. Corollary 9.39 of Unitary reflection groups by Lehrer and Taylor).

Etingof-Ginzburg's claim follows from this by decomposing $$\mathbf{C}[V \oplus V^*]=\mathbf{C}[V] \otimes \mathbf{C}[V^*]$$ and applying the above observations separately to each tensor factor.