MATH 7120 - Representation Theory of Real and p-adic Reductive Groups

Dan Barbasch, spring 2016.

This course will provide an introduction to the theory of infinite dimensional representations of reductive Lie groups. It will start with some basic results of Harish-Chandra on the structure of infinite dimensional representations, their characters as eigendistributions for invariant differential operators, and the fact that such groups are type I; this implies that the generalization of the Plancherel formula (from classical Fourier analysis) makes sense, and also that representations on Hilbert spaces admit a spectral decomposition according to irreducible representations. The course will continue with the classification of irreducible representations, following the work of Langlands, Vogan, Bernstein and many others. 

Prerequisites

A basic knowledge of Lie groups, Lie algebras and their structure.

References

Various papers of Harish Chandra to be found in his collected works.

A. Knapp, Representation theory of semisimple groups: An overview based on examples

A. Knapp, Lie groups beyond an Introduction

D. Vogan, Representations of real reductive Groups

D. Vogan, Cohomological Induction and Unitary Representations (with A. Knapp)

N. Wallach, Real reductive Groups vol I and II

W. Casselmann, Notes on representations of p-adic groups

J. Bernstein (and Rumelhart), Representations of p-adic groups, 1992 Harvard, lecture notes