Operator algebras and representation theory of Lie groups and groupoids – OpART

Operator algebras and representation theory share a common origin in the early twentieth century. Hermann Weyl was solving problems in both spectral geometry (eg, Weyl’s asymptotic formula) and representation theory (eg, the Peter-Weyl theorem) and the same can be said for Von Neumann, Gelfand, Dixmier, and many others. Recent advances are again showing the synergy between the two subjects. This project will use operator algebras coming from Lie groups and Lie groupoids to solve specific problems in representation theory and harmonic and microlocal analysis.
To cite an example of the influence of representation theory on operator algebras, consider the recent proof of the Helffer-Nourrigat conjecture, which proposes a representation theoretic criterion for the hypoellipticity of differential operators which are polynomials in vector fields satisfying Hörmander’s bracket-generating condition. Conversely, explicit constructions of unitary representations of real semisimple Lie groups and their intertwining operators can often be obtained from the analysis of (sub)elliptic differential and pseudodifferential operators, and can be generalized using techniques from equivariant index theory.
We aim to refine and generalize the above results to obtain classes of pseudo-differential and Fourier and integral operators which are adapted to novel geometric situations, including multifiltered manifolds. These operators can then be applied to better understand the structure of the unitary dual of a real semi-simple Lie group. Specifically, we will obtain refinements of the Baum-Connes and Connes-Kasparov conjectures, and also obtain explicit constructions of the branching operators which define the Howe correspondence between unitary representations of a dual reductive pair of groups.