Computational approaches to Poisson traces associated to finite subgroups of Sp(2n,C)

Abstract

We reduce the computation of Poisson traces on quotients of symplectic vector spaces by finite subgroups of symplectic automorphisms to a finite one, by proving several results which bound the degrees of such traces as well as the dimension in each degree. This applies more generally to traces on all polynomial functions which are invariant under invariant Hamiltonian flow. We implement these approaches by computer together with direct computation for infinite families of groups, focusing on complex reflection and abelian subgroups of GL(2,C) < Sp(4,C), Coxeter groups of rank <= 3 and A_4, B_4=C_4, and D_4, and subgroups of SL(2,C).