An Impossibility Theorem for Linear Symplectic Circle Quotients

Abstract

Abstract. We prove that when d > 2, a d-dimensional symplectic quotient at the zero level of a unitary circle representation V such that V S 1 = {0} cannot be Z-graded regularly symplectomorphic to the quotient of a unitary representations of a finite group.

Symplectic reduction at zero angular momentum

“…Our strategy for constructing maps between symplectic quotients is based on polynomial invariant theory and emphasizes the role of the (graded) R-algebra of regular functions R[M 0 ] on the symplectic quotient M 0 . This approach has been already advocated in [7,15,13,14]; the requisite material will be recalled in Section 2. Essential for the construction of the symplectomorphisms will be a system of real polynomials Q i,j in the invariants whose locus coincides with that of the moment map.…”

“…is as well normal by [31,Theorem 3.16], hence A C is integrally closed. Moreover, it can be seen as in the proof of [14,Theorem 4] that A C is a separating algebra in the sense of [6,Definition 2.3.8]. Specifically, the map (U × U * ) K /L → (U × U * )/H induced by the embedding (U × U * ) K → U × U * is injective as it is injective on the open dense set of points with isotropy group K and hence a birational map between normal varieties.…”

“…By a linear symplectic orbifold, we mean the symplectic orbifold given by the quotient of C n by a finite subgroup of U n . The argument below uses ingredients developed in [13,Section 3] and [14,Section 3]. Proof.…”

“…By [14,Theorem 2], we have that χ restricts to a Z + -graded regular symplectomorphism χ| S cl : S cl → T cl , where the global charts for these spaces are the restrictions of those for M 0,k,k and U/H, respectively. By Lemma 4.2, the map Tensoring with C yields isomorphisms…”

When is a symplectic quotient an orbifold?

“…Then as the other representations listed in Equation Remark 5.6. We note that the algebra isomorphisms inducing the Z-graded regular symplectomorphisms described in Propositions 5.3 and 5.5 can be computed explicitly, and were suggested by the observation that the Hilbert series coincide in [13]. In particular, the algorithm of Bedratyuk [3] yields the complex SL 2 -invariants.…”