Poisson Homology in Degree 0 for some Rings of Symplectic Invariants

Let $\go{g}$ be a finite-dimensional semi-simple Lie algebra, $\go{h}$ a Cartan subalgebra of $\go{g}$, and $W$ its Weyl group. The group $W$ acts diagonally on $V:=\go{h}\oplus\go{h}^*$, as well as on $\mathbb{C}[V]$. The purpose of this article is to study the Poisson homology of the algebra of invariants $\mathbb{C}[V]^W$ endowed with the standard symplectic bracket. To begin with, we give general results about the Poisson homology space in degree 0, denoted by $HP_0(\mathbb{C}[V]^W)$, in the case where $\go{g}$ is of type $B_n-C_n$ or $D_n$, results which support Alev's conjecture. Then we are focusing the interest on the particular cases of ranks 2 and 3, by computing the Poisson homology space in degree 0 in the cases where $\go{g}$ is of type $B_2$ ($\go{so}_5$), $D_2$ ($\go{so}_4$), then $B_3$ ($\go{so}_7$), and $D_3=A_3$ ($\go{so}_6\simeq\go{sl}_4$). In order to do this, we make use of a functional equation introduced by Y. Berest, P. Etingof and V. Ginzburg. We recover, by a different method, the result established by J. Alev and L. Foissy, according to which the dimension of $HP_0(\mathbb{C}[V]^W)$ equals 2 for $B_2$. Then we calculate the dimension of this space and we show that it is equal to 1 for $D_2$. We also calculate it for the rank 3 cases, we show that it is equal to 3 for $B_3-C_3$ and 1 for $D_3=A_3$.