Hans-Christian Herbig

An Impossibility Theorem for Linear Symplectic Circle Quotients

Using explicit computations of Hilbert series, 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.

WHEN IS A SYMPLECTIC QUOTIENT AN ORBIFOLD

Let K be a compact Lie group of positive dimension. We show that for most unitary K-modules the corresponding symplectic quotient is not regularly symplectomorphic to a linear symplectic orbifold (the quotient of a unitary module of a finite group). When K is connected, we show that even a symplectomorphism to a linear symplectic orbifold does not exist. Our results yield conditions that preclude the symplectic quotient of a Hamiltonian K-manifold from being locally isomorphic to an orbifold. As an application, we determine which unitary SU2-modules yield symplectic quotients that are Z+-graded regularly symplectomorphic to a linear symplectic orbifold. We similarly determine which unitary circle representations yield symplectic quotients that admit a regular diffeomorphism to a linear symplectic orbifold.

The Hilbert series and

Abstract Let V be a finite-dimensional representation of the complex circle C × determined by a weight vector a ∈ Z n . We study the Hilbert series Hilb a ( t ) of the graded algebra C [ V ] C a × of polynomial C × -invariants in terms of the weight vector a of the C × -action. In particular, we give explicit formulas for Hilb a ( t ) as well as the first four coefficients of the Laurent expansion of Hilb a ( t ) at t = 1 . The naive formulas for these coefficients have removable singularities when weights pairwise coincide. Identifying these cancelations, the Laurent coefficients are expressed using partial Schur polynomials that are independently symmetric in two sets of variables. We similarly give an explicit formula for the a-invariant of C [ V ] C a × in the case that this algebra is Gorenstein. As an application, we give methods to identify weight vectors with Gorenstein and non-Gorenstein invariant algebras.

a

This work concerns the moment map

Koszul properties of the moment map of some classical representations

\mu

The Koszul complex of a moment map

μ associated with the standard representation of a classical Lie algebra. For applications to deformation quantization it is desirable that

On compositions with x 2 /(1-x)

S/(\mu )