Research in Representation Theory

表示理论和不变量理论在数论、物理、组合学和几何中有着重要的应用。这个建议强调解析理论和代数理论。主要推力是广义惠特克模型的解析理论的扩展。其中包括贝塞尔模型，它在自同构形式和l -函数的研究中发挥了重要作用，这些模型源于将局部朗兰兹对应理论扩展到GL(n)内部扭曲之外的群的尝试。这些惠特克模型的重要性在于它们给出了多元自同构形式的傅里叶系数的表示理论解释。迄今为止，关于贝塞尔模型的数学文献只研究了退化主级数的子表示模型。这种类型的最一般的工作是PI？S关于以前的拨款。在本论文中，他建议对这些模型进行全面的分析。这项工作的一部分将与他的学生劳尔·戈麦斯合作进行。他希望在第一个夏天结束之前解决紧凑型稳定器的问题。一般情况下，将更加困难，并将强调在未来两年的资助（戈麦斯毕业后）。其他工作将涉及更多的不变理论中的代数几何问题，包括半单李代数中奇异集的结构和仿射变异上约化群作用的不变量环的Hilbert级数。这项工作如果成功，将影响几何和物理的应用。一般抽象。表示和对称在最近一些最深刻的数学问题的解决中发挥了关键作用。这些包括费马的证明？最后定理和三维庞卡罗猜想的证明。第一个例子使用表示理论来计算证明朗兰兹纲领的一个特例——谷山-志村猜想所需的局部因子。第二部分涉及研究黎曼曲率张量的各种不变量以及它们在里奇流下如何演变。本提案旨在扩展这些重大突破所涉及的表示理论，从而进一步发展数论（朗兰兹程序的更多案例），扩展里奇流和相关几何的适用性，应用于与特殊李群相关的共形场论的一些惊人的最新发展。

Representation theory and invariant theory have important applications to, in particular, number theory, physics, combinatorics and geometry. This proposal emphasizes both the analytic and the algebraic theory. The main thrust is an extension of the analytic theory of generalized Whittaker models. These include the Bessel models that have played an important role in the study of automorphic forms and L-functions stemming from attempts at extending the theory of the local Langlands correspondence to groups beyond inner twists of GL(n). The importance of these Whittaker models is that they give a representation theoretic interpretation of the Fourier coefficients of multivariate automorphic forms. The mathematical literature, to date, on Bessel models studies only models for subrepresentations of degenerate principal series. The most general work of this type is the PI?s on previous grants. In this grant he proposes to do a complete analysis of these models for generic principal series. The work will be in part carried out in collaboration with his student Raul Gomez. He expects to have solved the problem for the case of compact stabilizer before the end of the first summer. The general case will be much harder and will be emphasized in the next two years of the grant (after Gomez graduates). The other work will involve more algebraic geometric problems in invariant theory including the structure of the singular set in a semisimple Lie algebra and the Hilbert series of rings of invariants of actions of reductive groups on affine varieties. This work, if successful, will impact applications to geometry and physics.General Abstract. Representations and symmetry have played key roles in the recent solutions of some of the most profound problems in mathematics. These include the proof of Fermat?s Last Theorem and proof of The Poincaré Conjecture in three dimensions. The first example uses representation theory to calculate local factors needed in the proof of a special case of the Langlands Program known as the Taniyama-Shimura conjecture. The second involves the study of the various invariants of the Riemann Curvature tensor and how they evolve under the Ricci Flow. This proposal aims to extend the representation theory involved in these major breakthroughs leading to applications to further developments in number theory (more cases of the Langlands Program) , extensions of applicability of the Ricci flow and related geometry, applications to some amazing recent developments in Conformal Field theory related to exceptional Lie groups.