Poisson homology in degree 0 for some rings of symplectic invariants

Abstract

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 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 ( C [ V ] W ) , in the case where 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 g is of type B 2 ( so 5 ), D 2 ( so 4 ), then B 3 ( so 7 ), and D 3 = A 3 ( so 6 ≃ 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 ( 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 .