Compactly supported analytic indices for Lie groupoids

For any Lie groupoid we construct an analytic index morphism taking values in a modified $K-theory$ group which involves the convolution algebra of compactly supported smooth functions over the groupoid. The construction is performed by using the deformation algebra of smooth functions over the tangent groupoid constructed in \cite{Ca2}. This allows in particular to prove a more primitive version of the Connes-Skandalis Longitudinal index Theorem for foliations, that is, an index theorem taking values in a group which pairs with Cyclic cocycles. As other application, for $D$ a $\gr-$PDO elliptic operator with associated index $ind D\in K_0(\ci_c (\gr))$, we prove that the pairing $$<ind D,\tau>,$$ with $\tau$ a bounded continuous cyclic cocycle, only depends on the principal symbol 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's conjecture.