Intertwining Operators, Plancherel Measures and Automorphic L-functions

David GoldbergDMS-98 01340 Goldberg教授将在当地的朗兰兹项目中研究四个相关的问题。他继续研究关于某些经典群的某些非一般表示的Plancherd测度的朗兰兹猜想。他还将尝试确定附在拟分裂酉群的任意抛物子群上的标准交织算子的极点。此外，Goldberg教授还将研究p-ady域的特殊线性群的半单类型。最后，他将通过对R-群的系统研究来检验朗兰兹的另一个猜想。本世纪下半叶最重要的数学思想之一是，解析公式经常编码离散信息。例如，一个人可能想要计算一个特定方程的解的数量，但发现很难找到解。另一方面，您可能想要计算一系列方程的解的数量，并将答案视为一个序列。数学家称这类问题为“离散”问题。微积分中出现的函数并不是“离散的”，而是“分析的”，对这些函数来说，微积分运行得很好。令人惊讶的是，正确的解析函数可以包含上述离散例子的答案或答案。事实上，最早的例子是在几百年前发现的，但在过去的三十年里，这个学科已经系统化了。朗兰兹计划是一个有25年历史的项目，涉及到这一系统化。该项目是朗兰兹计划的重要组成部分。

David GoldbergDMS-98 01340Professor Goldberg will work on four related questions in the local Langlands program. He continue his work on Langlands conjecture about Plancherd measures for certain non-generic representations of some classical groups. He will also attempt to determine the poles of standard intertwining operators attached to arbitrary parabolic subgroups of quasisplit unitary groups. In addition, Professor Goldberg will study semisimple types of the special linear group of a p-adic field. Finally he will test another conjecture of Langlands through the systematic study of R-groups.One of the most important mathematical ideas of the second half of this century, is that analytic formula often encodes discrete information. For example, one might want to count the number of solutions of a particular equation, but discover that the solutions are very hard to find. On the other hand, you might want to count the number of solutions to a sequence of equations and consider the answers as a sequence. Mathematicians call these kind of problems 'discrete'. The functions that appear in calculus, and for which calculus works so well, are not 'discrete', but 'analytic.' Amazingly, the right kind of analytic function can contain the answer or answers to the discrete examples above. Actually the first examples of this were discovered a couple hundred years ago, but in the past thirty years, the subject has been systematized. The Langlands program is a 25 year old project involving this systematization. This project is an important part of the Langlands program.