The Hilbert Series of a Linear Symplectic Circle Quotient

We study the Hilbert series of the graded algebra of regular functions on a symplectic quotient of a unitary circle representation and elaborate explicit formulas for the lowest coefficients of the Laurent expansion of such a Hilbert series in terms of rational symmetric functions of the weights. Considerable efforts are devoted to including the cases where the weights are degenerate. We find that these Laurent expansions formally resemble Laurent expansions of Hilbert series of graded rings of real invariants of finite subgroups of Un. Moreover, we prove that certain Laurent coefficients are strictly positive. Experimental observations are presented concerning the behavior of the coefficients, and we provide empirical evidence that these results might generalize to higher dimensional tori and possibly nonabelian groups.