Dirac operators in representation theory

The mathematical structure that describes the symmetries that appear in nature is a simple algebraic object called a group. For example, one can consider the symmetries of a circle in the plane, i.e., length-preserving transformations of the plane that map the circle to itself. Rotations by any angle preserve the circle. But this set of symmetries has some intrinsic additional structure, e.g., performing one rotation followed by another gives another rotation in the same set and every rotation has an inverse rotation. This set, together with the additional structure, is called the special orthogonal group in the plane, and it is an example of a Lie group. Lie groups, named after the Norwegian mathematician Sophus Lie, are mathematical objects underlying the continuous symmetries inherent in a system.

This proposals falls in the area of representations of Lie groups. Representations are ways in which Lie groups can manifest themselves, e.g., rather than regarding the special orthogonal group as an abstract object, one can think of its `representation' as transformations of the plane given by rotations. The study of representations of Lie groups has a long and illustrious history and has had transformative impact in number theory and theoretical physics. The main idea of the present project is to import and generalize a beautiful mathematical construction, called the Dirac operator. The Dirac operator originated in physics by the famous work of Paul Dirac in quantum mechanics, and subsequently, found a home in mathematics (geometry) by the seminal work of Atiyah and Singer, and in the representation theory of Lie groups in the work of Parthasarathy, Atiyah-Schmid, Kostant, and many others. The new algebraic approach to the theory was initiated by Vogan about 15 years ago with the introduction of Dirac cohomology and this has opened new and exciting perspectives of research in mathematics. The current project will extend the Dirac theory to an algebraic setting and apply the techniques of the Dirac operator and Dirac cohomology to the world of representations of p-adic Lie groups and of related algebraic structures (Hecke algebras) with applications to modern number theory and areas of mathematical physics.

The proposal brings in a novel approach to algebraic representation theory, by exploring the applications of the Dirac operators to new algebraic settings. The proposal falls into an area of pure mathematics, called abstract harmonic analysis of Lie groups, that has been the subject of sustained and extraordinarily productive international research. The proposal will help to turn the UK into a leading place for research in this important mathematical field. The applications of the research in the proposal will have impact on modern number theory and on mathematical physics.

The results of the proposed research will be disseminated widely through talks at national and international conferences, graduate schools aimed at training postgraduate students and young researchers, at university seminars, and via papers published in top mathematical journals. The PI will also engage in outreach activities, for example, by giving public lectures targeted at senior secondary school students interested in mathematics and science.