Higher localized analytic indices and strict deformation quantization
Résumé
This paper is concerned with the localization of higher analytic indices for Lie groupoids. Let $\gr$ be a Lie groupoid with Lie algebroid $A\gr$. Let $\tau$ be a (periodic) cyclic cocycle over the convolution algebra $\cg$. We say that $\tau$ can be localized if there is a correspondence \begin{equation}\nonumber K^0(A^*\gr)\stackrel{Ind_{\tau}}{\longrightarrow}\mathbb{C} \end{equation} satisfying $Ind_{\tau}(a)=\langle ind\, D_a,\tau \rangle$ (Connes pairing). In this case, we call $Ind_{\tau}$ the higher localized index associated to $\tau$. In \cite{Ca4} we use the algebra of functions over the tangent groupoid introduced in \cite{Ca2}, which is in fact a strict deformation quantization of the Schwartz algebra $\sw(A\gr)$, to prove the following results: \begin{itemize} \item Every bounded continuous cyclic cocycle can be localized. \item If $\gr$ is {é}tale, every cyclic cocycle can be localized. \end{itemize} 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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