Mathematical Sciences: Research in Representation Theory andAutomorphic Forms

摘要 Wallach DMS-9531908 该项目将继续首席研究员的 约化群的表示论及其 应用于数论、几何和物理。之一 这个项目的主要主题涉及使用（所谓的）奇异酉表示的研究。 一种我们称之为转移的技术，它在相同的真实的形式之间"移动"酉表示 复杂化 要实现这一理念，必须 关于限制的非常详细的信息 （一般为非紧）对称表示 分组。 这种分析本身就很有意义 并已领导（并将领导）研究人员（包括PI） 豪的还原理论的惊人概括 对偶对到特殊群。 也会导致 一些值得注意的特殊情况的确认 B的照片。 Gross和D. 普拉萨德 另一个关键 本研究的方向是继续研究 不变多项式环的模理论 约化李代数上的微分算子 这 工作已经导致了一个新的解释， 斯普林格通信。 这里的新方向将 涉及某种"相对斯普林格对应"。 该项目还涉及与马修克莱格联合工作， 开发一个符号代数软件包（将被称为 "Groebner"）设计用于大规模计算， 并行超级计算机或分布式工作网络 站的 这个项目中描述的问题的解决方案将扩大我们对一些最令人兴奋的 发展的互动表示理论与 数学和物理学中看似无关的部分。 为 示例一个奇异表示的详细分析 (by Kostant和他的同事）已经导致了新的解决方案， 爱因斯坦的方程式 类型表示论 在这个项目中研究的也是一个重要的成分， 朗格兰的纲领中有一小部分是怀尔斯提出的 导致了他（怀尔）对费马大定理的证明。

Abstract Wallach DMS-9531908 This project will continue the principal investigator's research in the representation theory of reductive groups and its applications to number theory, geometry and physics. One of the main themes in this project involves the study of (so called) singular unitary representations using a technique we call transfer which "moves" unitary representations between real forms of the same complexification. To implement this idea one must have very detailed information on the restriction of representations to (generally non-compact) symmetric subgroups. This analysis is of interest in its own right and has led (and will lead) researchers (including the PI) to surprising generalizations of Howe's theory of reductive dual pairs to exceptional groups. It will also lead to confirmation of special cases of some remarkable conjectures of B. Gross and D. Prasad. Another key direction of this research involves the continued study of the module theory of the ring of invariant polynomial differential operators on a reductive Lie algebra. This work has already led to a new interpretation of the Springer correspondence. The new direction here will involve some sort of "relative Springer correspondence". This project also involves joint work with Matthew Clegg on the development of a symbolic algebra package (to be called "Groebner") designed for large scale computations using parallel supercomputers or distributed networks of work stations. The solution to the problems described in this project would expand our understanding of some of the most exciting developments in the interaction of representation theory with apparently unrelated parts of mathematics and physics. For example the detailed analysis of one singular representation (by Kostant and his coworkers) has led to new solutions to Einstein's equations. Representation theory of the type studied in this project is also an important ingredient in the Langland's program of which one very small part due to Wiles led to his (Wile's) proof of Fermat's Last Theorem.