Mathematical Sciences: Plancherel Measures and Automorphic L-Functions

本研究涉及表征理论和自同构形式以下领域的研究。首先研究了用轨道积分转移的映射对偶来定义p进约群的缓和l包的可能性。这张图的存在虽然尚未得到普遍证明，但现在已被普遍接受，它是朗兰兹稳定示踪公式的基础。第二个问题是关于由p进群的非离散缓变表示导出的r群的理论。首席研究员还将致力于统一研究自同构l函数的不同方法。最后，他将把他关于某些自同构L-函数极点有限性的结果推广到更大的一类自同构L-函数。算术函数的生成函数是研究算术函数的有力工具。在所有这些研究中，最深入、最博学的是所谓的朗兰纲领。但是这个项目的好处是非常高的。这项研究就在这个领域。

This research concerns the study of the following areas of representation theory and automorphic forms. The first is the study of the possibility of defining tempered L-packets for p- adic reductive groups by means of the dual of a certain map coming from transfer of orbital integrals. The existence of this map even though not yet proved in general is by now fairly accepted and is the basis of stabilization of the trace formula by Langlands. The second problem concerns the theory of R-groups for representations induced from non-discrete tempered representations of a p-adic group. The Principal Investigator will also pursue his work on unifying different methods of studying automorphic L-functions. Finally he will try to extend his result on the finiteness of poles of certain automorphic L- functions to a larger class of them. Generating functions of arithmetic functions are well known and powerful tools for the study of these functions. The deepest and most erudite of all these studies is the so called Langland's program. But the benefits of this program are very high. This research is in this area.