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: 0.56). 01/2006; 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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