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, \(K\)-finite, smooth and analytic vectors, admissible representations, unitary representations, the Harish-Chandra admissibility and analyticity theorems, \((\mathfrak{g},K)\)-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 \((\mathfrak{g},K)\)-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 \(SL_2(\Bbb R)\) 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 \(G\) 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 \(\mathcal O\) of representations of a complex semisimple Lie algebra \(\mathfrak{g}\). These representations do not lift to the corresponding complex Lie group \(G\) in general, but we will see that they are related to representations of \(G\) which are in general non-holomorphic and infinite-dimensional. Namely, we will introduce the notion of a Harish-Chandra bimodule over \(\mathfrak{g}\), and explain that the study of such bimodules is equivalent, in a suitable sense, to the study of non-holomorphic representations of \(G\) (this is just the above correspondence between representations of a semisimple real Lie group \(G\) and \((\mathfrak{g},K)\)-modules in the special case when \(G\) 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 \(\mathcal{O}\). 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 \(SL_2(\Bbb C)\).

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 \(G\) (originally due to Langlands) in terms of \(K_{\Bbb C}\)-orbits on the flag variety \(G_{\Bbb C}/B\). Since this material is rather technical, we skip many proofs in this part of the course.