Computational Approaches to Poisson Traces Associated to Finite Subgroups of ${\rm Sp}_2(\mathbb{C})$

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 that bound the degrees of such traces as well as the dimension in each degree. This applies more generally to traces on all polynomial functions that 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 ${\rm GL}_2(\mathbb{C}) \lt {\rm Sp}_4(\mathbb{C})$, Coxeter groups of rank $\le 3$ and types $A_4$, $B_4 = C_4$, and $D_4$, and subgroups of ${\rm SL}_2(\mathbb{C})$.