Le groupe des traces de Poisson de la variete quotient h+h*/W en rang 2

Details Le groupe des traces de Poisson de la variete quotient h+h*/W en rang 2 Le groupe des traces de Poisson de la variete quotient h+h*/W en rang 2

2006-03-06

arxiv.org/abs/math/0603142v2

Mathematics

Rings and Algebras

24 pages, french, revised version. Accepted for publication in Communications in Algebra in a shorter version

Scientific paper

Let $V$ be a symplectic space over $\mathbb{C}$, $dim_\mathbb{C} V=2l$, and let $G$ be a finite subgroup of $Sp(V)$. The invariant regular functions $\mathbb{C}[V]^G$ inherit a Poisson algebra structure and so the quotient variety ${\cal X}=V/G$ becomes then an affine algebraic Poisson variety. One can now consider the non commutative deformation of $\cal X$ given by the invariant algebra $A_l(\mathbb{C})^G$, where $A_l(\mathbb{C})$ stands for the Weyl algebra of rank $l$. There exist two families of natural examples of this situation. The first concerns wreath products of a finite subgroup of $SL(2,\mathbb{C})$ with an appropriate symmetric group acting on $(\mathbb{C}^2)^n$; the second family is constructed with a Weyl group $W$ acting on the double of the reflexion representation ${\mathfrak{h}}\oplus {\mathfrak{h}}^*$. A nice result of Berest, Etingof and Ginzburg establishes the finiteness of the dimension of $HP_0({\cal X})= \mathbb{C}[\cal X]/\{\mathbb{C}[\cal X], \mathbb{C}[\cal X]\}$, the Poisson trace group of $\cal X$. The purpose of this work is to compute this dimension in certain cases and in particular to compare it to the dimension of the usual trace group of the above mentioned non commutative deformation. The principal theorem establihed here is : {\bf Theorem.} With the above notations, we have the following equality: $$dim_\mathbb{C} HP_0({{\mathfrak{h}}\oplus {\mathfrak{h}}^*}/W)=dim_\mathbb{C} HH_0(A_l(\mathbb{C})^W).$$ Moreover, this common dimension is 1 in type $ A_2$, 2 in type $B_2$ and 3 in type $G_2$. We also give examples where the difference of these two dimensions is unbounded.

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