Representation Theory of p-Adic Groups

摘要 菲利普·库茨科爱荷华州大学DMS-97 32527 Kutzko教授将使用类型方法研究约化p进群的光滑复表示理论。 群中的类型是紧开子群加上不可约表示，使得该对携带关于所有不可约表示的信息。 这种方法在过去已经成功地用于研究其他线性群。 在其他项目中，研究者将尝试应用类型的方法，应用于分裂经典群的背景下，以理解抛物线诱导表示的还原。 世纪后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，你可能想计算一个方程序列的解的数量，并将答案视为一个序列。数学家称这类问题为“离散”问题。出现在微积分中的函数，以及微积分如此有效的函数，不是“离散的”，而是“分析的”。令人惊讶的是，正确的解析函数可以包含上面离散例子的答案。 实际上，第一个这样的例子是在几百年前发现的，但在过去的30年里，这个主题已经系统化，成为数论的一个分支，称为朗兰兹纲领。 该项目是朗兰兹计划的重要组成部分。

ABSTRACT Philip Kutzko University of Iowa DMS-97 32527 Professor Kutzko will study the smooth complex representation theory of a reductive p-adic group using the method of types. A type in a group is a compact open subgroup together with an irreducible representation in such a way that the pair carries information about all irreducible representations. This approach has been used successfully in the past to study other linear groups. Among other projects, the investigator will try to apply the method of types be applied in the context of split classical groups with the goal of understanding reducibility of parabolically induced representations. One of the most important mathematical ideas of the second half of the 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 into a branch of number theory called the Langlands program. This project is an important part of the Langlands program.