This project will explore new connections between the emerging area of noncommutative geometry and the representation theory of Lie groups. The first part of the project was inspired initially by explorations in mathematical physics. It revives a proposal of George Mackey to correspond representations of a semisimple Lie group with those of its Cartan motion group, but it does so in the light of more recent developments in noncommutative geometry. The second component of the project seeks to develop links between the index-theoretic approach to representation theory incorporated into the Baum-Connes theory and the geometric representation theory of Beilinson and Bernstein. A third segment of the project aims to investigate more deeply the connection between the Baum-Connes theory and the Langlands classification of irreducible representations. The goal of the final major portion of the project is to frame the "quantization commutes with reduction" phenomenon in symplectic geometry within noncommutative geometry. It is expected that this will lead to a clearer understanding of the range of the phenomenon. Although the project initially involves the relatively well-understood representation theory of compact groups, a long-term aim of the project is to apply insights gained more broadly within representation theory, guided by the outlooks of the individual components.
Group representation theory is a recurring theme in modern mathematics. Its origins lie within algebra, but the subject has important ties to geometry, to differential equations, and to many other mathematical areas. This project focuses on the groups that capture mathematically the concept of continuous symmetry (such as the continuous, rotational symmetry of a circle, which may be rotated about itself by any angle, as opposed to the discrete symmetry of a square, which may be rotated about itself only by quarter turns). These groups are basic to the mathematical expression of the laws of physics, thanks to the continuous symmetries (rotations, translations, and others) intrinsic to physical space and time. The fundamental observable quantities in physical science such as energy and momentum are paired with these symmetries. For example, the law of conservation of energy is a restatement of the expectation that the laws of physics remain unchanged as time passes. In quantum theory, the symmetries of space and time imply that fundamental particles correspond to so-called irreducible unitary group representations, and it becomes a matter of interest and importance to determine these representations. Within mathematics, the same representations have been found to be intimately linked to the theory of special functions, to number theory, and to differential equations, among other areas. It is the common objective of the various components of this project to bring new mathematical techniques to bear on representation theory. The long-term goal is to conceptualize and deepen our understanding of an area that has been central to mathematics and its applications for more than a hundred years.
