Research in Representation Theory and Automorphic Forms

本项目将研究表示论、非交换调和分析和代数群论中的几个相关问题。 在表示论中，它涉及真实的约化群的小酉表示的构造、研究和应用。 在调和分析中，它涉及到在一个真实的约化群上快速减少函数的Paley-Wiener定理中放弃K-有限条件。 这个分析将被应用于证明一个版本的Casselman-Wallach定理取决于参数。 这个定理将隐含非K-有限Eisenstein级数的亚纯延拓。 与这些更分析的问题，我们将研究代数问题，确定分次多重性的不可约成分的行动，约化代数群的仿射锥。 后者的工作也有应用程序的分析和研究的措施纠缠在quantum computing.Representation理论有其根源在19世纪不变理论，早期的20世纪量子力学和中期的20世纪数论。 在二十一世纪的第一个十年里，这一理论又回到了它的根源。 世纪的不变理论强调具体问题的二进制形式与算法的解决方案。这些问题再次出现，现在正在推广到量子计算。 早期的量子力学研究了光子、电子等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在量子信息技术中重新出现。 希尔伯特空间方法也催生了表示论，它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，即建立一个非对易类场论（怀尔对费马大定理的证明实际上是对Tanayama-Shimura猜想的一个特例的证明，而Tanayama-Shimura猜想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.