  Let $V$ be a symplectic space over $\mathbb{C}$, $dim_\mathbb{C} V=2l$, and
let $G$ be a finite subgroup of $Sp(V)$. The invariant regular functions
$\mathbb{C}[V]^G$ inherit a Poisson algebra structure and so the quotient
variety ${\cal X}=V/G$ becomes then an affine algebraic Poisson variety. One
can now consider the non commutative deformation of $\cal X$ given by the
invariant algebra $A_l(\mathbb{C})^G$, where $A_l(\mathbb{C})$ stands for the
Weyl algebra of rank $l$. There exist two families of natural examples of this
situation. The first concerns wreath products of a finite subgroup of
$SL(2,\mathbb{C})$ with an appropriate symmetric group acting on
$(\mathbb{C}^2)^n$; the second family is constructed with a Weyl group $W$
acting on the double of the reflexion representation ${\mathfrak{h}}\oplus
{\mathfrak{h}}^*$. A nice result of Berest, Etingof and Ginzburg establishes
the finiteness of the dimension of $HP_0({\cal X})= \mathbb{C}[\cal
X]/\{\mathbb{C}[\cal X], \mathbb{C}[\cal X]\}$, the Poisson trace group of
$\cal X$. The purpose of this work is to compute this dimension in certain
cases and in particular to compare it to the dimension of the usual trace group
of the above mentioned non commutative deformation. The principal theorem
establihed here is : {\bf Theorem.} With the above notations, we have the
following equality: $$dim_\mathbb{C} HP_0({{\mathfrak{h}}\oplus
{\mathfrak{h}}^*}/W)=dim_\mathbb{C} HH_0(A_l(\mathbb{C})^W).$$ Moreover, this
common dimension is 1 in type $ A_2$, 2 in type $B_2$ and 3 in type $G_2$. We
also give examples where the difference of these two dimensions is unbounded.
