Abstract
This paper is concerned with the localization of higher analytic indices for Lie groupoids. Let G be a Lie groupoid with Lie algebroid AG. Let τ be a (periodic) cyclic cocycle over the convolution algebra \( C_c^\infty \left( G \right) \) We say that τ can be localized if there is a morphism
satisfying Ind τ (a)=〈ind D a, τ 〉 (Connes pairing). In this case, we call Ind τ the higher localized index associated to τ. In [CR08a] we use the algebra of functions over the tangent groupoid introduced in [CR08b], which is in fact a strict deformation quantization of the Schwartz algebra S(AG ), to prove the following results:
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Every bounded continuous cyclic cocycle can be localized.
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If G is étale, every cyclic cocycle can be localized.
We will recall this results with the difference that in this paper, a formula for higher localized indices will be given in terms of an asymptotic limit of a pairing at the level of the deformation algebra mentioned above. We will discuss how the higher index formulas of Connes-Moscovici, Gorokhovsky-Lott fit in this unifying setting.
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Rouse, P.C. (2010). Higher localized analytic indices and strict deformation quantization. In: Abbaspour, H., Marcolli, M., Tradler, T. (eds) Deformation Spaces. Vieweg+Teubner. https://doi.org/10.1007/978-3-8348-9680-3_4
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