Résume | Poisson traces are linear functionals on a Poisson algebra which annihilate Poisson brackets. They are dual to the zeroth Poisson (or Lie) homology, and the dimension of this space bounds the number of irreducible finite-dimensional representations of any quantization. I will consider the case where the Poisson algebra is the algebra of $G$-invariant polynomials in $2n$ variables with complex coefficients, where $G$ is a finite subgroup of $Sp(2n,C)$. I will prove some results which reduce the computation of Poisson traces, for fixed $n$ and $G$, to a finite one that can be done by computer, and will describe several results of this computation. I will also classify the complex reflection groups $G$ for $n = 2$ for which the dimension of the Poisson traces coincides with the (well-known) dimension of the zeroth Hochschild homology of the algebra of $G$-invariant differential operators in n variables, which a priori is only a lower bound. The results also imply this equality for Coxeter groups of rank $\leq 3$ and the Weyl groups $B_4$ and $D_4$. These results also give the Hilbert series of the space of Poisson traces, in terms of the polynomial degree (and when the aforementioned equality holds, also this series for the zeroth Hochschild homology). This is joint work with Etingof, Gong, Pacchiano, and Ren, part of which took place in the context of undergraduate research at MIT. |