Research in Representation Theory and Automorphic Forms

本课题将研究表示理论、非交换调和分析和代数群论中的几个相关问题。在表示理论中，它涉及实约化群的小酉表示的构造、研究和应用。在调和分析中，丢掉了实约化群上速降函数的Paley-Wiener定理中的k -有限条件。这个分析将被应用于证明卡塞尔曼-瓦拉赫定理的一个依赖于参数的版本。这个定理将暗示非k有限爱森斯坦级数的亚纯延拓。与这些更解析的问题相关，我们将研究确定仿射锥上可约代数群作用的不可约成分的梯度复数的代数问题。后一项工作也适用于量子计算中纠缠度量的分析和研究。表征理论起源于19世纪的不变量理论、20世纪早期的量子力学和20世纪中期的数论。在21世纪的第一个十年里，这一理论又回到了它的根源。19世纪的不变量理论强调用算法解决二进制形式的具体问题。这些问题再次出现，现在被推广应用于量子计算。早期的量子力学研究的是涉及光子、电子等的令人困惑和奇怪的测量。这些现象导致了量子力学的希尔伯特空间方法。早期量子力学的哲学争论作为量子信息技术重新出现。希尔伯特空间方法也产生了表征理论，这是它在数论中的主要应用之一。朗兰兹纲领为21世纪确立了一个目标，即建立一个非交换类场论（威尔对费马大定理的证明实际上是对田山-志村猜想的一个特例的证明，这是朗兰兹纲领的一个特例）。这个项目是所有这些令人兴奋的方向的接口。

AbstractWallachThis project will study several related problems in representation theory, non-commutative harmonic analysis and algebraic group theory. In representation theory it involves the construction, study and application of small unitary representations of real reductive groups. In harmonic analysis, it involves dropping the K-finite condition in Paley-Wiener theorems for rapidly decreasing functions on a real reductive group. This analysis will be applied to proving a version of the Casselman-Wallach theorem depending on parameters. This theorem will imply a meromorphic continuation of non-K-finite Eisenstein series. Related to these more analytic problems we will study the algebraic problem of determining graded multiplicities for the irreducible constituents of the action of a reductive algebraic group on an affine cone. The latter work also has applications to the analysis and to the study of measures of entanglement in quantum computing.Representation theory has its roots in nineteenth century invariant theory, early twentieth century quantum mechanics and mid-twentieth century number theory. In this first decade of the twenty first century the theory has returned to its roots. The nineteenth century invariant theory emphasized concrete questions on binary forms with algorithmic solutions. These problems have reemerged and are now being generalized to apply to quantum computation. Early quantum mechanics studied puzzling and weird measurements involving photons, electrons etc. These phenomena led to the Hilbert space approach to quantum mechanics. The philosophical debates of the early quantum mechanics have reemerged as quantum information technology. The Hilbert space approach also gave birth to representation theory, which has as one of its main applications in number theory. The Langlands program has established a goal for the twenty first century to establish a non-commutative class field theory (Wile's proof of Fermat's Last Theorem is actually proof of a special case of the Tanayama-Shimura conjecture which is a special case of the Langlands program). This project is in the interface of all of these exciting directions.