Mathematical Sciences: Research in Representation Theory andAutomorphic Forms

本项目将继续主要研究者对约化群的表示理论及其在数论、几何和物理中的应用的研究。这个项目的主要主题之一涉及到（所谓的）奇异酉表示的研究，使用一种我们称之为转移的技术，在相同复杂程度的真实形式之间“移动”酉表示。为了实现这个想法，必须有非常详细的信息来限制表示（通常是非紧致的）对称子群。这种分析本身就很有趣，并且已经（并将继续）引导研究人员（包括PI）将Howe的约化对偶理论惊人地推广到特殊群体。这也将导致对B. Gross和D. Prasad的一些非凡猜想的特殊案例的确认。本研究的另一个重点方向是继续研究约化李代数上不变多项式微分算子环的模理论。这项工作已经导致了对施普林格对应的新解释。这里的新方向将涉及某种“相对施普林格对应”。该项目还包括与Matthew Clegg共同开发符号代数包（称为“Groebner”），该包专为使用并行超级计算机或分布式工作站网络进行大规模计算而设计。这个项目中所描述的问题的解决方案将扩展我们对表征理论与数学和物理学中明显不相关的部分相互作用中一些最令人兴奋的发展的理解。例如，对一个奇异表示的详细分析（由Kostant和他的同事）导致了爱因斯坦方程的新解。在这个项目中研究的类型的表示理论也是朗兰计划的一个重要组成部分，其中很小一部分由于怀尔斯导致了他（怀尔斯）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.