Abstract:

The theory of (infinite dimensional) representations of real reductive Lie groups is now almost forty years old, and its principal results as the Plancherel formula or the classification of irreducible representations may be regarded as "classical". More recently, Beilinson and Bernstein discovered the "localization for modules over the enveloping algebra" which can be viewed as a generalization of the classical Borel-Weil-Bott theory. This relates representation theory to the geometry of flag manifolds and has lead to tremendous advances, most notably the determination of the characters of all irreducible representations. Looking at the resulting formulas, Vogan noticed that there also is an intimate connection with the geometry of the flag manifold of the Langlands dual group. Although this connection (together with its conjectural generalization to p-adic groups) is of compelling beauty, it remains a complete mystery at the moment: The proof (in the known cases) is just heavy combinatorics.

In this AG we want to focus on the approach to representation theory by localization and use it to study the structure of our representations in some detail. In particular we want to understand the Langlands classification from a geometric point of view. The classics and mysteries will be left to the first and last day respectively.