  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.
