Index theory and groupoids for filtered manifolds

In this thesis, we propose to use generalized fixed point algebras as an approach to the pseudodifferential calculus on filtered manifolds. A filtered manifold is a manifold \(M\) with a filtration of its tangent bundle which is compatible with the Lie bracket. This filtration allows to define a new notion of order for the differential operators on \(M\). As a result, the highest order part of a differential operator is a family of right-invariant model operators acting on certain nilpotent Lie groups. These groups form the bundle of osculating groups of the filtered manifold. The new order can be encoded by a dilation action of \(\mathbb{R}_{>0}\) on this bundle. The tangent groupoid of a filtered manifold \(M\) describes the relation between the operators acting on \(M\) and their model operators on the osculating groups. It is equipped with a "zoom" action of \(\mathbb{R}_{>0}\) that extends the dilations. In this thesis, we build the generalized fixed point algebra for the zoom action on a certain ideal \(J\) in the groupoid \(C^*\)-algebra of the tangent groupoid. This generalized fixed point algebra \(\mathrm{Fix}^{\mathbb{R}_{>0}}(J)\) is a \(C^*\)-subalgebra of the bounded operators on \(L^2(M)\). Moreover, there is a "principal symbol map" which takes values in another generalized fixed point algebra \(\mathrm{Fix}^{\mathbb{R}_{>0}}(J_0)\), where \(J_0\) is an ideal in the \(C^*\)-algebra of the bundle of osculating groups. The kernel of the symbol map consists of the compact operators. The symbol algebra is, in general, noncommutative. It is unital if \(M\) is compact. In this case, call \(P\in\mathrm{Fix}^{\mathbb{R}_{>0}}(J)\) elliptic if its principal symbol is invertible. We show that \(P\) is elliptic if and only if all model operators satisfy the Rockland condition. Furthermore, it is shown that the sequence above coincides with the \(C^*\)-completion of the order zero pseudodifferential extension by van Erp and Yuncken. When viewing a graded Lie group as a filtered manifold, we ...