Character Theory of Non-Linear Groups

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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.