C. R. Acad. Sci. Paris, t. 325, Sbrie I, p. 193-198, 1997 CbomCtrie diff&entielle/Differenfia/ Geometry Indice analytique et groupoiIdes de Lie Bertrand MONTHUBERT et Franqois PIERROT Institut de MathGmatiquas, Universitk P’lerre-et-Mario-Curie, (case 191) Tour 46-O. 3” itag?, 4. place Jussieu, 75232 Paris CEDEX 05, Franrr. E-mail : monthuhe@math.jussieu.fr pierrotQmath.jussku.fr RCsum6. Abstract. Soit G un groupoyde de Lie. Le calcul pseudodiff&entiel sur G dkfinit l’indice analytique.Le groupoi’de tangentassock! G induit un morphisme de K-tlkorie. Nous montrons que ce dernier coincide avec I’indice analytique. Ce dernier peut done se dkfinir sans recours au calcul pseudodiffkrentiel. Analytic index and Lie groupoids Let G be a Lie groupoid. The pseudodifferential calculus over G de&es the analytic inde.x. The tangent group&d induces a morphism in K-theory; we prove that it coincides with the analytic index. Thus the latter may be defined without considering the pseudodifSerentia1 calculus. A bridged English Version NOTATIONS. - Let G be a Lie groupoid, possibly with boundary, provided aG(‘) is saturated and Gi)G(O)= dG. Let TF be the normal fibre bundle of G(O) into G, and S*F be the associated sphere bundle. Let C,“(G) be the involutive algebra of functions with compact support over G, the product being given by convolution (which depends on a Haar system). This algebra can be canonically defined with the help of half-densities (see [3]). We have the canonical morphism U, : Cy(G,(?,) -+ C,“(G,(,)). 1. Pseudodifferential calculus over Lie groupoids We generalize the construction of A. Connes, built in the case of the holonomy groupoid of a foliation (see [2]). DEFINITION 1.1. - We call G-operator an operator T over C,?(G) such that there is a family T,, : Ccm(GtL) -+ Gm(G,) satisfying T(f)(y) = Ts:,c,)(fl~,,,,)(r), and such that U,-%(,)U, = ?1,(,). Note pdsentbe par Alain CONNES. 07644442/97/03250193 0 Acadbmie des ScienceslElsevier, Paris 193