Representations of Quantizations: Categorifications, Highest Weight Modules, and Harish-Chandra Bimodules

Representation theory is concerned with the study of linear symmetries. These symmetries form algebras that often arise as quantizations or more general deformations of functions on classical spaces -- this is a mathematical counterpart of passing from Classical to Quantum Mechanics or Field theory in Physics. Individual representations form categories, and understanding these categories leads to understanding the representations themselves. Tools to study categories include understanding their own symmetries and showing that categories of different origin are, in fact, equivalent. The focus of this project is to study categories of representations of quantizations, the symmetries of these categories, and their equivalences. The project also involves training young mathematicians and writing books and survey articles to benefit undergraduate and graduate students. In more detail, the project consists of three parts. The first part proposes the study of categories of highest weight modules over affine quantum algebras. These categories should be viewed as categorical analogs of the polynomial representations of double affine Hecke algebras. They depend on parameters, and the task is to establish derived equivalences between the categories corresponding to different parameters and to relate their t-structures. The second part of the project seeks to relate several categorical versions of the elliptic Hall algebra, a close relative of the double affine Hecke algebra, proving derived equivalences between these versions. The third part seeks a conceptual geometric understanding of unitary representations of complex semisimple Lie groups. 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.