Topological Methods in Representation Theory and Automorphic Forms

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9803862 Vilonen This research program consists of three separate, but intimately related, projects. The first one, joint with Wilfried Schmid, concerns one of the important remaining questions in representation theory of reductive Lie groups: determining all of their unitary representations. They will attack this problem using the geometric techniques they developed to prove the Barbasch-Vogan conjecture. The second project, joint with E.Frenkel, D.Gaitsgory, and D.Kazhdan, deals with the Langlands conjectures. They will continue their work on the geometric Langlands conjecture and the geometric theory of Whittaker functions. The purpose of the third project, joint with Mirkovic, is to develop geometric techniques for representation theory of algebraic groups. This project has also turned out to be important for work on the geometric Langlands conjecture. The goal is to apply their methods to representation theory of algebraic groups in positive characteristic as well as over the integers. One of the most interesting aspects of mathematics is to discover and understand connections between objects that from the outset seem unrelated. A very deep such connection has been proposed by Langlands and has become known as the Langlands correspondence. A fundamental theme of this research program is to establish such connections by using geometry. The situation is most interesting when the objects themselves are not geometric but the connection between them is established by geometric means. The joint work with Wilfried Schmid on the Barbasch-Vogan conjecture has this flavour. The other two projects of this research program are directly related to the Langlands correspondence. ***