Article

# The Guillemin-Sternberg conjecture for noncompact groups and spaces

Journal of K-theory K-theory and its Applications to Algebra Geometry and Topology (Impact Factor: 0.75). 01/2006; DOI: 10.1017/is008001002jkt022

Source: arXiv

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**ABSTRACT:**We first introduce an invariant index for G-equivariant elliptic differential operators on a locally compact manifold M admitting a proper cocompact action of a locally compact group G. It generalizes the Kawasaki index for orbifolds to the case of proper cocompact actions. Our invariant index is used to show that an analog of the Guillemin–Sternberg geometric quantization conjecture holds if M is symplectic with a Hamiltonian action of G that is proper and cocompact. This essentially solves a conjecture of Hochs and Landsman.Advances in Mathematics 01/2010; · 1.37 Impact Factor -
##### Article: Quantizing tame actions

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**ABSTRACT:**The quantization commutes with reduction problem was solved for compact groups acting on compact symplectic manifolds by Meinrenken and others in the late 1990s. Since then, generalizations have been obtained to cases where only the group, or the orbit space of the action, is compact. The main result in this paper is a generalization to settings where the group, the manifold and the orbit space may all be noncompact, as long as the symplectic reduction at zero is compact. The method used builds on Tian and Zhang's approach in the compact case. It works under a geometric assumption on the group action, called tameness. We show that several relevant classes of Hamiltonian actions are tame, such as cocompact actions, actions by subgroups on strongly elliptic coadjoint orbits, and actions by Lie groups on their cotangent bundles. The result obtained in this way leads to a version of the shifting trick, which motivates the definition of semi-formal quantization. This takes values in the K-homology of the reduced C*-algebra of the Lie group acting, which is a common generalization of the generalized representation ring of a compact group, and the K-theory of the reduced C*-algebra of a general group. Under certain conditions, this K-homological quantization reduces to earlier definitions for compact groups or compact orbit spaces. In the Kahler setting, the methods used yield Morse-type inequalities for a deformed Dolbeault complex.09/2013; - [Show abstract] [Hide abstract]

**ABSTRACT:**Groupoids generalize groups, spaces, group actions, and equivalence relations. This last aspect dominates in noncommutative geometry, where groupoids provide the basic tool to desingularize pathological quotient spaces. In physics, however, the main role of groupoids is to provide a unified description of internal and external symmetries. What is shared by noncommutative geometry and physics is the importance of Connes's idea of associating a C-algebra C(Gamma) to a Lie groupoid Gamma: in noncommutative geometry C(Gamma) replaces a given singular quotient space by an appropriate noncommutative space, whereas in physics it gives the algebra of observables of a quantum system whose symmetries are encoded by Gamma. Moreover, Connes's map Gamma|-->C(Gamma) has a classical analogue Gamma|-->A(Gamma) in symplectic geometry due to Weinstein, which defines the Poisson manifold of the corresponding classical system as the dual of the so-called Lie algebroid A(Gamma) of the Lie groupoid Gamma, an object generalizing both Lie algebras and tangent bundles. Only a handful of physicists appear to be familiar with Lie groupoids and Lie algebroids, whereas the latter are practically unknown even to mathematicians working in noncommutative geometry: so much the worse for its relationship with symplectic geometry! Thus the aim of this review paper is to explain the relevance of both objects to both audiences. We do so by outlining the road from canonical quantization to Lie groupoids and Lie algebroids via Mackey's imprimitivity theorem and its symplectic counterpart. This will also lead the reader into symplectic groupoids, which define a `classical' category on which quantization may speculatively be defined as a functor into the category KK defined by Kasparov's bivariant K-theory of C-algebras. This functor unifies deformation quantization and geometric quantization, the conjectural functoriality of quantization counting the ``quantization commutes with reduction'' conjecture of Guillemin and Sternberg among its many consequences.Journal of Geometry and Physics 01/2006; 56:24-54. · 1.06 Impact Factor

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