Research in Representation Theory and Automorphic Forms

本项目将研究表示论、非对易调和分析和代数群论中的几个相关问题。在表示论中，它涉及实约化群的小么正表示的构造、研究和应用。在调和分析中，它涉及到去掉实约群上快速递减函数的Paley-Wiener定理中的K-有限条件。这一分析将被应用于证明依赖于参数的Casselman-Wallach定理的一个版本。这个定理意味着非K-有限Eisenstein级数的亚纯延拓。与这些更具分析性的问题相关，我们将研究确定可约代数群在仿射锥上作用的不可约分量的分次重数的代数问题。表象理论起源于十九世纪不变量理论、二十世纪早期的量子力学和二十世纪中叶的数论。在21世纪的头十年里，这一理论又回到了它的根基。十九世纪的不变量理论强调关于二进制形式的具体问题和算法解决方案。这些问题已经重新出现，现在正被推广应用于量子计算。早期的量子力学研究涉及光子、电子等令人费解和奇怪的测量。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以量子信息技术的形式重新出现。希尔伯特空间方法也催生了表象理论，它是数论中的主要应用之一。朗兰兹计划为二十一世纪确立了建立非对易类场论的目标(Wile对Fermat最后定理的证明实际上是Tanayama-Shimura猜想的一个特例的证明，它是朗兰兹计划的一个特例)。这个项目是所有这些令人兴奋的方向的交汇点。

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.