On pseudodifferential operators on filtered and multifiltered manifolds

We present various different approaches to constructing algebras of pseudodifferential operators adapted to particular geometric situations. A general goal is the study of index problems in situations where standard elliptic theory is insufficient. We also present some applications of these constructions.We begin by presenting a characterization of pseudodifferential operators in terms of distributions on the tangent groupoid which are essentially homogeneous with respect to the natural $\mathbb{R}^\times_+$-action. This is carried out in the generality of filtered manifolds, which are manifolds that are locally modelled on nilpotent groups, generalizing contact, CR and parabolic geometries. We describe the tangent groupoid of a filtered manifold, and use this to construct a pseudodifferential calculus analogous to the unpublished calculus of Melin.Next, we describe a rudimentary multifiltered pseudodifferential theory on the full flag manifold X of a complex semisimple Lie group G which allows us to simultaneously treat longitudinal pseudodifferential operators along every one of the canonical fibrations of X over smaller flag manifolds. The motivating application is the construction of a G-equivariant K-homology class from the Bernstein-Gelfand-Gelfand complex of a semisimple group. This construction been completely resolved for only a few groups, and we will discuss the remaining obstacles as well as the successes.Finally, we discuss pseudodifferential operators on two classes of quantum flag manifolds. First, we consider quantum projective spaces, where we can generalize the abstract pseudodifferential theory of Connes and Moscovici to obtain a twisted algebra of pseudodifferential operators associated to the Dolbeault-Dirac operator. Secondly, we look at the full flag manifolds of $SU_ q(n)$, where we instead generalize the multifiltered construction of the classical flag manifolds, thus obtaining an equivariant fundamental class for the full flag variety of $SU_q(3)$ from the Bernstein-Gelfand-Gelfand complex. As applications, we obtain equivariant Poincaré duality of the quantum flag manifold of $SU_q(3)$ and the Baum-Connes Conjecture for the discrete dual of $SU_q(3)$ with trivial coefficients