Syllabus

Course Meeting Times

Lectures: 2 sessions / week, 80 minutes / session

Prerequisites

18.745 Lie Groups and Lie Algebras I or 18.755 Lie Groups and Lie Algebras II

Course Overview

The goal of this course is to give an introduction to the representation theory of non-compact semisimple Lie groups. It will rely on some material from 18.745 Lie Groups and Lie Algebras I and 18.755 Lie Groups and Lie Algebras II described in the Full Lecture Notes: Lie Groups and Lie Algebras I & II (PDF) (which, in particular, develop the representation theory of compact Lie groups). However, full familiarity with this material is not required.

The two most essential differences between representation theories of compact and non-compact Lie groups are that irreducible representations of non-compact Lie groups are in general infinite-dimensional, and that representations are not completely reducible. This makes the story much more complicated than in the compact case. Namely, the first aspect necessitates a non-trivial input from functional analysis (as representations are realized in Fréchet and Hilbert spaces), while the second aspect requires bringing in category theory and homological algebra.

We will begin with a review of the relevant functional analysis, including Fréchet spaces, continuous representations, algebras of measures on a Lie group, %28K%29-finite, smooth and analytic vectors, admissible representations, unitary representations, the Harish-Chandra admissibility and analyticity theorems, %28(\mathfrak{g},K)%29-modules, infinitesimal equivalence and realizations, Harish-Chandra’s globalization theorem. The main takeaway of this part is that the most essential structural features of representation theory of non-compact semisimple Lie groups are captured by %28(\mathfrak{g},K)%29-modules (or Harish-Chandra modules), which are certain algebraic skeletons of representations. The rest of the course is purely algebraic and entails a detailed study of the category of such modules. At the end of this part we use this theory to classify irreducible representations of %28SL_2(\Bbb R)%29 and determine which of them are unitary.

Then we pass to the main topic of the course, which is a detailed study of representations of a complex semisimple Lie group %28G%29 regarded as a real group (i.e., not necessarily holomorphic representations). We begin with developing the necessary algebraic machinery (the Chevalley restriction theorem, the Chevalley-Shepard-Todd theorem, Kostant’s theorem, Harish-Chandra isomorphism).

Next we discuss the Bernstein-Gelfand-Gelfand category %28\mathcal O%29 of representations of a complex semisimple Lie algebra %28\mathfrak{g}%29. These representations do not lift to the corresponding complex Lie group %28G%29 in general, but we will see that they are related to representations of %28G%29 which are in general non-holomorphic and infinite-dimensional. Namely, we will introduce the notion of a Harish-Chandra bimodule over %28\mathfrak{g}%29, and explain that the study of such bimodules is equivalent, in a suitable sense, to the study of non-holomorphic representations of %28G%29 (this is just the above correspondence between representations of a semisimple real Lie group %28G%29 and %28(\mathfrak{g},K)%29-modules in the special case when %28G%29 is a complex group regarded as a real group). We will also show that the structure of Harish-Chandra bimodules is closely related to the structure of category %28\mathcal{O}%29. Using this connection, we will classify irreducible Harish-Chandra bimodules. Along the way, we will obtain a number of other important results, such as the structure of the nilpotent cone, the Duflo-Joseph theorem, and Duflo’s primitive ideal theorem. Finally, as the most basic example of this theory, we will completely describe the irreducible representations of %28SL_2(\Bbb C)%29.

The last part of the course is an introduction to geometric representation theory of real semisimple Lie groups. Here we discuss the Borel-Weil theorem, the Springer resolution of the nilpotent cone, and its quantization—the Beilinson-Bernstein localization theorem. Then we briefly review the theory of D-modules and describe the D-module approach to representation theory of real semisimple Lie groups based on the Beilinson-Bernstein localization. In particular, we describe the classification of irreducible representations of such a group %28G%29 (originally due to Langlands) in terms of %28K_{\Bbb C}%29-orbits on the flag variety %28G_{\Bbb C}/B%29. Since this material is rather technical, we skip many proofs in this part of the course.