Poisson homology in degree 0 for some rings of symplectic invariants

Let g be a finite-dimensional semi-simple Lie algebra, h a Cartan subalgebra of g, and W its Weyl group. The group W acts diagonally on V:=h⊕h∗, as well as on C[V]. The purpose of this article is to study the Poisson homology of the algebra of invariants CW[V] endowed with the standard symplectic bracket.To begin with, we give general results about the Poisson homology space in degree 0, denoted by HP0(CW[V]), in the case where g is of type Bn−Cn or Dn, 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 g is of type B2 (so5), D2 (so4), then B3 (so7), and D3=A3 (so6≃sl4). 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 HP0(CW[V]) equals 2 for B2. Then we calculate the dimension of this space and we show that it is equal to 1 for D2. We also calculate it for the rank 3 cases, we show that it is equal to 3 for B3−C3 and 1 for D3=A3.