On pseudodifferential operators on filtered and multifiltered manifolds

TL;DR

This paper develops new pseudodifferential operator algebras for filtered and multifiltered manifolds, including quantum analogues, to address index problems beyond standard elliptic theory, with applications in representation theory and quantum geometry.

Contribution

It introduces novel constructions of pseudodifferential operators on filtered and multifiltered manifolds, including quantum cases, and applies them to index theory and equivariant K-homology.

Findings

Characterization of pseudodifferential operators via tangent groupoid distributions.

Development of multifiltered pseudodifferential theory on flag manifolds.

Construction of equivariant K-homology classes from BGG complexes on quantum flag manifolds.

Abstract

This memoir is a summary of recent work, including collaborations with Erik van Erp, Christian Voigt and Marco Matassa, compiled for the "Habilitation \`a diriger des recherches". We present various different approaches to constructing algebras of pseudodifferential operators adapted to filtered and multifiltered manifolds and some quantum analogues. 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 on filtered manifolds in terms of distributions on the tangent groupoid which are essentially homogeneous with respect to the natural $\mathbb{R}^\times_+$-action. Next, we describe a rudimentary multifiltered pseudodifferential theory on the full flag manifold $\mathcal{X}$ of a complex semisimple Lie group $G$ which allows us to simultaneously treat longitudinal pseudodifferential operators along every one of the canonical fibrations of $\mathcal{X}$ 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. Finally, we discuss pseudodifferential operators on two classes of quantum flag manifolds: quantum projective spaces and the full flag manifolds of $SU_q(n)$. In particular, on the full flag variety of $SU_q(3)$ we obtain an equivariant fundamental class from the Bernstein-Gelfand-Gelfand complex.