In this thesis, we study singular pseudo-differential operators defined by groupoids satisfying the Lauter-Nistor condition, by a method parallel to that of manifolds with boundary and edge differential operators. The example of the Bruhat sphere is studied in detail. In particular, we construct an extension to the calculus of uniformly supported pseudo-differential operators that is analogous to the calculus with bounds defined on manifolds with boundary. We derive a Fredholmness criterion for operators on the Bruhat sphere, and prove that their parametrices up to compact operators lie inside the extended calculus; we construct the heat kernel of perturbed Laplacian operators; and prove an Atiyah-Singer type renormalized index formula for perturbed Dirac operators on the Bruhat sphere using the heat kernel method.