Collaborative Research: Rank and Duality in Representation Theory

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This project concerns representation theory. Symmetry is a familiar concept in elementary geometry -- many of the most important figures (lines, circles, squares) are symmetrical. It is less widely appreciated that symmetry has been found to be fundamental for understanding the world. Both the theory of relativity and quantum mechanics, two major developments in physics during the 20th century, rely heavily on ideas of symmetry. Linear algebra is another mathematical development of the late 19th and 20th centuries that is now heavily used throughout science. Representation theory is the study of how symmetry can be combined with linear algebra. This project deals with correspondences, called eta correspondences, between systems of symmetries of different objects. The first phase of this project is to show that there are eta correspondences for some of the most important finite systems of symmetries, and how to describe these correspondences. Subsequent phases of the project will extend eta correspondences to more cases, refine the concepts used to describe them, and use them to apply representation theory to a broad range of questions in pure and applied mathematics.In more detail, this project introduces an innovative approach to the study of representations of classical groups over finite and local fields, an approach that seems beneficial for harmonic analysis. An effective theory of "size" for representations will be developed, including a precise definition and a method to analyze representations of a given size. The motivation in the finite setting comes from the fact that many questions about finite groups (e.g., random walks, word maps, Cayley graphs, etc.) can be approached using harmonic analysis. More precisely, what intervenes in such problems are the character ratios (character divided by dimension) of the irreducible representations (irreps) of the relevant group G. In general, it is not feasible to compute the character ratios exactly, but for applications it often suffices to show that the character ratios are small for most representations. Since in many cases the dimension of the representation is what makes the character ratio small, the first phase is to understand the dimensions of irreps and, especially, those with dimensions that are much smaller than average, since they most likely to make the dominant contributions to any sum of character ratios. The investigators have a theory that is applicable to all classical groups and, perhaps, even to all reductive groups over finite and local fields. They propose several different notions of rank of a representation, and they suspect that, although different in nature, these notions are equivalent. Having these notions in hand gives a lot of information on the dimensions of the irreps of G. In addition, the investigators discovered a systematic construction, called the eta correspondence, between large naturally defined families of irreps of G of a given rank, and (all, or most of) the irreps of a smaller group H. There is reason to believe that this construction is exhaustive, and the project pursues a proof of this conjecture. The eta correspondence gives strong control over character ratios for the representations it constructs, and a formal treatment of this relation will form the second phase of the project. A significant discovery so far is that although the dimensions of irreps of a given rank vary considerably, the character ratios of these irreps are nearly equal. Thus, for purposes of harmonic analysis, representations of a fixed rank form a natural family to study. Finally, in the third phase of the project, the investigators will apply bounds on character ratios and dimensions to several open problems in group theory and its applications.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.