NSF-BSF Research in Representation Theory with Applications to Number Theory and Physics

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Representation theory is the study of linear symmetries. Since there are many symmetries in nature, and since linear methods are powerful and efficient, this subject has found numerous applications in mathematics, physics, chemistry, and computer science. Classically, representations of finite or compact groups were studied. The current project concerns representations of infinite, non-compact, algebraic groups, and the PI and his collaborator will study more recent mathematical structures such as Lie supergroups, quantum groups, and double affine Hecke algebras. Each of these topics has been an active area of modern mathematical research for many years, yet many important questions remain open. The resolution of the questions answered in this project will lead to significant new applications in number theory, combinatorics, and physics.The main part of the project is dedicated to the study of automorphic representations and will be pursued by the PI and an Israeli collaborator D. Gourevitch. This is a subject of central importance in the Langlands program in number theory, and recently of much interest in string theory. The PI and Gourevitch will employ the method of Whittaker functionals, which are a traditional tool for studying large automorphic representations. However the PI and collaborators have recently introduced a more refined class of functionals, which are well-adapted to studying small representations. This will have applications to quantum gravity, where small automorphic representations arise as certain quantum corrections to Einstein's theory of general relativity. The second part of the project will study higher dimensional analogs of Dirichlet series, which generalize the celebrated Riemann zeta function. The PI will study these series using double affine Hecke algebra, which are a traditional tool for understanding the symmetries of Macdonald polynomials -- themselves a central object of study in algebraic combinatorics. The PI and collaborators have recently discovered a new class of representations of these algebras, called metaplectic representations, that hold considerable promise for studying multiple Dirichlet series. The third part of the project will develop the theory of Capelli differential operators. These operators have their origin in classical invariant theory, which comprised a large part of 19th century mathematics. They also played a key role in Atiyah-Bott-Patodi approach to the index theorem, which is a highlight of 20th century mathematics. The PI and collaborators will extend Capelli operators to the modern realms of Lie superalgebras and quantum 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.