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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 SU 2 -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. [ABSTRACT FROM AUTHOR]

Abstract: We develop differential and symplectic geometry of differentiable Deligne–Mumford stacks (orbifolds) including Hamiltonian group actions and symplectic reduction. As an application we construct new examples of symplectic toric DM stacks. [Copyright &y& Elsevier]

Abstract: With an eye towards index theoretic applications we describe a Schubert like stratification on the Grassmannian of hermitian lagrangian spaces in . This is a natural compactification of the space of hermitian matrices. The closures of the strata define integral cycles, and we investigate their intersection theoretic properties. We achieve this by blending Morse theoretic ideas, with techniques from o-minimal (or tame) geometry and geometric integration theory. [Copyright &y& Elsevier]

Abstract: Using the analytic assembly map that appears in the Baum–Connes conjecture in noncommutative geometry, we generalise the Guillemin–Sternberg conjecture that ‘quantisation commutes with reduction’ to (discrete series representations of) semisimple groups G with maximal compact subgroups K acting cocompactly on symplectic manifolds. We prove this generalised statement in cases where the image of the momentum map in question lies in the set of strongly elliptic elements , the set of elements of with compact stabilisers. This assumption on the image of the momentum map is equivalent to the assumption that , for a compact Hamiltonian K-manifold N. The proof comes down to a reduction to the compact case. This reduction is based on a ‘quantisation commutes with induction’-principle, and involves a notion of induction of Hamiltonian group actions. This principle, in turn, is based on a version of the naturality of the assembly map for the inclusion . [Copyright &y& Elsevier]