Unipotent Representations and Associated Cycles

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Laws of nature, in particularly those of physics, satisfy basic symmetries. In mathematical language, these symmetries are most often expressed in terms of Lie groups and their representations (named after Sophus Lie, a 19th century mathematician who pioneered these concepts in his study of solutions of differential equations). This project is concerned with properties of unitary representations. The modern study of unitary representations stems from quantum physics. In addition, they play a crucial role in many other applications such as tomography, crystallography, and signal processing.In more technical terms, this project is focused on properties of unipotent representations, which are the building blocks of the unitary duals of real and p-adic reductive groups. The determination of the unitary dual is a major problem in the representation theory of such groups. The PI will formulate and sharpen conjectures on the signatures of hermitian forms for modules of the affine Hecke algebra. These are essential for the determination of the unitary duals of p-adic groups. Another focus of this project is to make explicit the role of unipotent representations in the description of the unitary dual. In addition to linear groups, the PI will study these notions for nonlinear and disconnected groups that arise in mathematical physics and in the study of automorphic forms.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.