Let G be a locally compact group acting smoothly and properly by isometrieson a complete Riemannian manifold M, with compact quotient. There is anassembly map which associates to any G-equivariant K-homology class on M, anelement of the topological K-theory of a suitable Banach completion B of theconvolution algebra of continuous compactly supported functions on G. The aimof this paper is to calculate the composition of the assembly map with theChern character in the entire cyclic homology of B. We prove an index theoremreducing this computation to a cup-product in bivariant entire cycliccohomology. As a consequence we obtain an explicit localization formula whichincludes, as particular cases, the equivariant Atiyah-Segal-Singer indextheorem when G is compact, and the Connes-Moscovici index theorem forG-coverings when G is discrete. The proof is based on the bivariant Cherncharacter introduced in previous papers.
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