Unitary Representations of Real Reductive Groups and Mixed Hodge Modules

Representation theory can be broadly understood as the mathematical study of symmetry. The symmetries that arise in quantum mechanics are called "unitary representations." One of the major unsolved problems in representation theory is to classify unitary representations. This problem has driven a considerable portion of all research in representation theory over the past eighty years; its solution would have far-reaching ramifications in several neighboring fields, including number theory, harmonic analysis, signal processing, and theoretical physics. The PI proposes to develop a new geometric approach to this problem. This approach will lead to substantial new insights into the structure of unitary representations and, hopefully, a solution to the problem of computing the unitary dual. The project also provides research training opportunities for graduate students. There are two main existing approaches to the study of unitary representations: the orbit method philosophy of Kirillov and Kostant, and the Hodge theory approach of Schmid and Vilonen. The orbit method seeks to parameterize the unitary dual of a Lie group G in terms of (roughly speaking) orbits for G on the dual space of its Lie algebra. The Hodge theory approach seeks to understand unitary representations by localizing over the flag variety and applying tools from Hodge theory. These approaches work along two different axes. Whereas the orbit method provides mainly hints as to where one should look for unitary representations, Hodge theory provides a powerful apparatus for proving that representations are unitary. Neither approach seems sufficient, on its own, to solve the problem of computing the unitary dual. In this project the PI proposes to develop a unified geometric approach to unitary representation theory by combining the orbit method with techniques from Hodge theory. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.