An impossibility theorem for linear symplectic circle quotients (Q887028)

In their earlier paper [Exp. Math. 23, No. 1, 46--65 (2014; Zbl 1301.53091)], the authors computed the Hilbert series of the graded algebra of regular functions on a symplectic quotient of a unitary circle representation. Actually, they derived there many explicit formulas for the lowest coefficients of the Laurent expansion of such a Hilbert series in terms of rational symmetric functions of the weights. In the present paper they use the results of these computations in order to compare the symplectic quotients of the unitary circle representations with some finite unitary quotients. The main result of the paper (Theorem 1) states that if \(S^1 \longrightarrow U(V)\) is a unitary circle representation such that \(V^{S^1}\equiv {0}\) and the dimension of the symplectic quotient \(M_0\) is greater than \(2\), then there can not exist a \({\mathbb Z}\)-graded symplectomorphism to a quotient of \({\mathbb C}^n\) by a finite subgroup \(\Gamma < U_n= U({\mathbb C}^n)\).