Article

# The Guillemin-Sternberg conjecture for noncompact groups and spaces

(Impact Factor: 0.69). 01/2006; 1(03). DOI: 10.1017/is008001002jkt022
Source: arXiv

ABSTRACT

The Guillemin-Sternberg conjecture states that "quantisation commutes with
reduction" in a specific technical setting. So far, this conjecture has almost
exclusively been stated and proved for compact Lie groups \$G\$ acting on compact
symplectic manifolds, and, largely due to the use of spin_c Dirac operator
techniques, has reached a high degree of perfection under these compactness
assumptions. In this paper we formulate an appropriate Guillemin-Sternberg
conjecture in the general case, under the main assumptions that the Lie group
action is proper and cocompact. This formulation is motivated by our
interpretation of the "quantisation commuates with reduction" phenomenon as a
special case of the functoriality of quantisation, and uses equivariant
K-homology and the K-theory of the group C*-algebra C*(G) in a crucial way. For
example, the equivariant index - which in the compact case takes values in the
representation ring R(G) - is replaced by the analytic assembly map - which
takes values in K_0(C*(G)) - familiar from the Baum-Connes conjecture in
noncommutative geometry. Under the usual freeness assumption on the action, we
prove our conjecture for all Lie groups G having a cocompact discrete normal
subgroup, but we believe it is valid for all unimodular Lie groups.

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• "Let D LG denote the corresponding Spin c -Dirac operator on M G . Following [9], we make the assumption that the quotient space M/G is compact, that is, the G-action on M is cocompact. Then M G = µ −1 (0)/G is also compact. "
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• "This approach is indeed successful, cf. [5] [6]. Our plan is to extend the approach another step further. "
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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.
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