Poisson Homology in Degree 0 for some Rings of Symplectic Invariants

TL;DR

This paper investigates the degree 0 Poisson homology of invariant rings under Weyl group actions for certain Lie algebra types, providing explicit calculations and supporting conjectures in symplectic invariant theory.

Contribution

It offers new calculations of Poisson homology in degree 0 for specific Lie algebra types and ranks, confirming and extending previous results and conjectures.

Findings

Dimension of $HP_0$ is 2 for $B_2$

Dimension of $HP_0$ is 1 for $D_2$

Dimension of $HP_0$ is 3 for $B_3$ and 1 for $D_3$

Abstract

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 $D_3=A_3$.