Research in Representation Theory

表示论和不变量理论在数论、物理学、组合数学和几何学中有重要的应用。这个建议强调了分析和代数理论。主要推力是广义惠特克模型的分析理论的扩展。其中包括贝塞尔模型，它在自守形式和L-函数的研究中发挥了重要作用，这些自守形式和L-函数源于试图将局部朗兰兹对应理论扩展到GL（n）内部扭曲之外的群。 这些惠特克模型的重要性在于，它们给出了多元自守形式的傅立叶系数的表示论解释。数学文献，到目前为止，贝塞尔模型的研究只模型的子表示退化的主要系列。这种类型的最一般的工作是PI？s在以前的赠款。在这个补助金，他建议做一个完整的分析，这些模型的通用主要系列。这项工作将部分与他的学生劳尔·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。一般情况下将更难，并将在未来两年的赠款（戈麦斯毕业后）强调。其他工作将涉及更多的代数几何问题的不变量理论，包括结构的奇异集在一个半单李代数和希尔伯特系列环的不变量的行动约化群仿射品种。这项工作，如果成功的话，将影响几何和物理的应用。表示和对称性在最近一些最深刻的数学问题的解决方案中发挥了关键作用。包括费马的证明？在三维空间中证明了庞加莱猜想。第一个例子使用表示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。第二部分研究了黎曼曲率张量的各种不变量以及它们在里奇流下的演化。这个建议的目的是扩展涉及这些重大突破的表示理论，从而应用于数论的进一步发展（朗兰兹纲领的更多案例），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.