Mathematical Sciences: Research in Representation Theory andAutomorphic Forms

Wallach DMS-9531908这个项目将继续主要研究者在约化群表示理论及其在数论、几何和物理中的应用方面的研究。这个项目的主要主题之一涉及使用一种我们称为转移的技术来研究(所谓的)奇异么正表示，该技术在具有相同复杂性的实形式之间“移动”酉表示。要实现这一思想，必须有关于(通常是非紧的)对称子群的表示的限制的非常详细的信息。这一分析本身就很有趣，并且已经(也将会引导)研究人员(包括PI)令人惊讶地将豪的还原对偶理论推广到例外群。这也将导致B.Gross和D.Prasad的一些值得注意的猜想的特例得到证实。这项研究的另一个重要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斯普林格通信的新解释。这里的新方向将涉及到某种形式的“斯普林格亲属通信”。该项目还涉及与马修·克莱格合作开发一个符号代数包(将被称为“Groebner”)，旨在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。这个项目中描述的问题的解决方案将扩大我们对表征理论与显然无关的数学和物理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(由科斯坦特和他的同事)导致了爱因斯坦方程式的新解。在这个项目中研究的这种类型的表示理论也是朗兰德程序中的一个重要组成部分，其中由于威尔斯的一小部分导致了他(Wile)对费马最后定理的证明。

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.