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For any Lie groupoid we construct an analytic index morphism taking values ina modified $K-theory$ group which involves the convolution algebra of compactlysupported smooth functions over the groupoid. The construction is performed byusing the deformation algebra of smooth functions over the tangent groupoidconstructed in \cite{Ca2}. This allows in particular to prove a more primitiveversion of the Connes-Skandalis Longitudinal index Theorem for foliations, thatis, an index theorem taking values in a group which pairs with Cyclic cocycles.As other application, for $D$ a $\gr-$PDO elliptic operator with associatedindex $ind D\in K_0(\ci_c (\gr))$, we prove that the pairing $$<ind D,\tau>,$$with $\tau$ a bounded continuous cyclic cocycle, only depends on the principalsymbol class $[\sigma(D)]\in K^0(A^*\gr)$. The result is completely general for{\'E}tale groupoids. We discuss some potential applications to the Novikov'sconjecture.

Compactly supported analytic indices for Lie groupoids