Character Theory of Non-Linear Groups

【摘要】线性约化群的表示理论，如GL(n)，是一个非常活跃的课题，部分原因是它在自同构形式和朗兰兹规划中的应用。然而，自同构形式的重要有趣的例子来自非线性群，即群p，它不能被实现为GL(n)的子群。实约群理论中的一个基本结果是Vogan对偶。在与Peter Trapa的合作中，我计划将其扩展到非线性群体。在这种情况下，有许多有趣的特征与线性情况大不相同。在与Rebecca Herb的一个联合项目中，我提议通过“提升”理论将实约非线性群与线性群联系起来，研究它们的特征。这种提升与这些群体的沃根对偶密切相关。在与Niranjan Ramachandran合作的一个项目中，我正在研究非线性p进约化群的结构。这是研究这些群体性格理论的重要的第一步。最后，安妮格雷特·保罗和我正在做一个项目来尽可能完整地参数化一维表示中R上的对应关系。

AbstractAdamsThe representation theory of linear reductive groups, such as GL(n) is a very active subject, in part because of applications to the automorphic forms and the Langlands program. However important interesting examples of automorphic forms come from non-linear groups, i.e. groups pwhich can not be realized as a subgroup of GL(n).A fundamental result in the theory of real reductive groups is Vogan duality. In joint work with Peter Trapa I plan to extend this to non-linear groups. There are a number of interesting features in this setting which are quite different from the linear case.In a joint project with Rebecca Herb I propose to study the characters of real reductive non-linear groups, by relating them to linear groups via a "lifting" theory. This lifting is closely related to Vogan duality for these groups.In a project with Niranjan Ramachandran I am studying the structure of non-linear p-adic reductive groups. This is an important first step in studying character theory for these groups.Finally Annegret Paul and I are working on a project to parametrize the theta-correspondence over R for one-dimensional representations as completely as possible.