Poisson Homology in Degree $0$ for some Rings of Symplectic Invariants
Résumé
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 $D_3=A_3$.
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