Representation Theory for Reductive Groups Over Local Fields

这位研究者建议研究一些与局部域上约化群的表示理论和调和分析有关的相关课题。有四个一般性的主题，其中一些被进一步划分。第一个主题涉及一个奇怪的新公式，它涉及厄米特对称情况下实群的离散级数表示的特征，这个公式无疑是比较Lefschetz公式和Arthur-Selberg迹公式所必需的。第二个主题涉及各种背景下的转移因素：内部形式、下村变种、扭曲转移因素的下降。第三个话题与下村品种的不良减产有关。目前，这方面的优先事项较低，但在未来三年内，这种情况可能会改变。第四个主题包括关于非阿基米德局部域上约化群的轨道积分的五个不同但相关的问题。研究人员建议用不那么专业的语言来研究一些属于自同构形理论的主题，自同形理论是一个美丽的数学领域，与数学的三个主要分支：代数、分析和几何联系在一起。对于这一领域的工作者来说，过去的几十年是令人兴奋的时期，其中一个亮点是在Wiles的壮观的费马最后定理的证明中，自同构形式的基本使用。未来几十年有望有许多令人兴奋的进一步发展，研究人员希望通过解决提案中提出的一些问题，以及通过帮助培训这个技术要求高的领域的年轻工人，间接地为这些发展做出直接贡献。

The investigator proposes to study a number of interrelated topics having to do with representation theory and harmonic analysis for reductive groups over local fields. There are four general topics, some of which are divided up further. The first topic concerns a strange new formula involving characters of discrete series representations of real groups in the Hermitian symmetric case, a formula that will no doubt be needed in order to compare the Lefschetz formula with the Arthur-Selberg trace formula. The second topic concerns transfer factors in various settings: inner forms, Shimura varieties, descent for twisted transfer factors. The third topic concerns bad reduction of Shimura varieties. Currently this has lower priority but that could change in the course of the next three years. The fourth topic consists of five distinct but related questions about orbital integrals for reductive groups over non-archimedean local fields. In less technical language the investigator proposes to study a number of topics that belong to the theory of automorphic forms, a beautiful area of mathematics with ties to all three main branches of mathematics: algebra, analysis and geometry. The last several decades have been exciting times for workers in this area, one of the highlights being the essential use of automorphic forms in Wiles's spectacular proof of Fermat's Last Theorem. The coming decades promise many further exciting developments, and the investigator hopes to contribute to these directly, by solving some of the questions raised in the proposal, as well as indirectly, by helping to train young workers in this technically demanding field.