Topological Methods in Representation Theory and Automorphic Forms

Northwestern University, Evanston IL

Investigators

Abstract

DMS-0105256 Kari Vilonen The goal of the the work of the principal investigator is to develop geometric techniques for applications in representation theory and the geometric Langlands program. From this point of view the basic geometric problem is to understand perverse sheaves from the microlocal point of view. A sufficiently deep understanding of microlocal perverse sheaves on flag manifolds should lead to interesting invariants of irreducible representations of reductive Lie groups and, hopefully, in the end, to an understanding of unipotent representations. Similar techniques are also being applied to modular representation theory. The goal of the last project is to gain insight in the geometric Langlands conjecture in its various forms. In addition to microlocal techniques the point of view of Whittaker models is used. The fundamental theme of the project is the interplay between geometry and representation theory. The idea of reducing the understanding of various geometries to their groups of symmetries goes back to Felix Klein at the end of the nineteenth century. From this point of view the basic objects, the "atoms" of the theory, are the irreducible representations of the group of symmetries of a given geometry. It turns out that these atoms arise from a particular geometric situation and hence we can apply geometric methods to analyze the properties of the irreducible representations. This is the goal of the first part of the the proposal. The second part of the project deals with the deep conjectures of Langlands. Here the geometry in question arises from an arithmetic situation and this brings number theory in the mixture of geometry and representation theory.