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

表征理论是对线性对称性的研究。由于自然界中有许多对称性，并且线性方法强大而有效，因此这门学科在数学、物理、化学和计算机科学中得到了许多应用。经典地，研究有限群或紧群的表示。目前的项目涉及无限的、非紧的代数群的表示，PI和他的合作者将研究更多最新的数学结构，如李超群、量子群和双仿射Hecke代数。这些主题多年来一直是现代数学研究的活跃领域，但许多重要问题仍未解决。在这个项目中回答的问题的解决将导致在数论，组合学和物理学中重要的新应用。该项目的主要部分是致力于自同构表示的研究，并将由PI和以色列合作者D. Gourevitch进行。这是朗兰兹数论计划中一个非常重要的课题，最近在弦理论中也引起了很大的兴趣。PI和Gourevitch将采用Whittaker泛函方法，这是研究大型自同构表示的传统工具。然而，PI和合作者最近引入了一种更精细的函数类，它非常适合研究小表示。这将应用于量子引力，其中小的自同构表示作为爱因斯坦广义相对论的某些量子修正而出现。项目的第二部分将研究狄利克雷级数的高维类比，它推广了著名的黎曼ζ函数。PI将使用双仿射Hecke代数来研究这些级数，这是理解麦克唐纳多项式对称性的传统工具-它们本身是代数组合学研究的中心对象。PI及其合作者最近发现了这些代数的一类新的表示，称为元表示，它对研究多重狄利克雷级数具有相当大的前景。项目的第三部分将发展Capelli微分算子的理论。这些算子起源于经典的不变量理论，它构成了19世纪数学的很大一部分。他们还在指数定理的atiyah - bot - patodi方法中发挥了关键作用，指数定理是20世纪数学的一个亮点。PI和合作者将把Capelli算子扩展到李超代数和量子群的现代领域。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.