Lie Groups and Representations: Mathematics GR6344 (Spring 2023)

Tuesday and Thursday, 10:10am - 11:25am
507 Mathematics

This course will cover various more advanced aspects of the theory of Lie groups, Lie algebras and their representations, from a point of view emphasizing relations to quantum mechanics and number theory.  It is aimed at first-year mathematics graduate students although students in physics might also find it of interest.  It is a continuation of the fall semester course taught by John Morgan.

Topics to be covered will likely include:

Heisenberg group, Stone-von Neumann theorem, some quantum mechanics
Relation to theta functions
The oscillator representation
Analogy with the spinor representation

Symplectic geometry and the orbit method

Review of universal enveloping algebra, classification theory of complex Lie algebras
Verma modules and highest-weight representations
Harish-Chandra homomorphism
Lie algebra cohomology and the Borel-Weil-Bott theorem (algebraic)
Classification of finite-dimensional representations of semi-simple complex Lie algebras

Geometric representation theory.  Borel-Weil-Bott theorem

Real semi-simple Lie groups and Lie algebras: classification

SL(2,R) and its representations, relation to modular forms


Tuesday, January 17:  Overview of course, review of material from first semester.  Notes
Thursday, January 19:  The Heisenberg group and its representations (part 1).  Notes
Tuesday, January 24:  The Heisenberg group and its representations (part 2). 
Thursday, January 26:  More on polarizations.  The symplectic group as automorphisms of the Heisenberg group
Tuesday, January 31:   The Poisson bracket. Lie algebra of the symplectic group. The oscillator representation. Notes
Thursday, February 2:  Analogy with spin groups and the spinor representation.  Notes
Tuesday, February 7:  Howe duality and the theta correspondence
Thursday, February 9:  Heisenberg and metaplectic groups over other fields
Tuesday, February 14:  Theta functions and automorphic forms
Thursday, February 16:  Review of classification theory of Lie algebras, the universal enveloping algebra
Tuesday, February 21:  Classification theory of semi-simple complex Lie algebras
Thursday, February 23:  Verma modules and highest-weight representations
Tuesday, February 28: Class canceled due to illness
Thursday, March 2:  More on Verma modules and highest-weight representations
Tuesday, March 7:  Infinitesimal character, Harish-Chandra homomorphism
Thursday, March 9:  Lie algebra cohomology: definition
Spring Break
Tuesday, March 21:  Lie algebra cohomology: cohomology of compact groups, Kostant's theorem
Thursday, March 23:  Lie algebra cohomology:  the Weyl character formula
Tuesday, March 28:  The orbit method and geometric quantization
Thursday, March 30:  Geometric representation theory.  Borel-Weil theorem
Tuesday, April 4:  Borel-Weil-Bott theorem
Thursday, April 6: Real semi-simple Lie groups and Lie algebras: classification
Tuesday, April 11:
Thursday, April 13: SL(2,R) and its representations
Tuesday, April 18:
Thursday, April 20:
Tuesday, April 25:  Relation to modular forms
Thursday, April 27:

Problem Sets, Exam

There will be a problem set due roughly every other week, and a take-home final exam. 

Problem set 1: due Tuesday, January 31.

Problem set 2
: due Tuesday, February 14.

Problem set 3: due Tuesday, March 7.

Problem set 4: due Tuesday, March 28.

Problem set 5: due Tuesday, April 11.

Problem set 6: due Tuesday,  April 25.



Some notes about background for this course:

Lie groups and Lie algebras
Representation theory
Simple quantum mechanical systems

Some sources for much of the material from the first semester that we'll be using are:

John Morgan's web-page here has lecture notes from the first semester.

Eckhard Meinrenken's lecture notes on Lie Groups and Lie Algebras.

Alexander Kirillov, Jr.
An Introduction to Lie Groups and Lie Algebras
Cambridge University Press, 2008
Note that the electronic version of this book is available freely for Columbia students at the link above or via its entry in the Columbia library catalog.
Much of the first seven chapters in this book were covered during the first semester. Later on in the course we'll cover what is in chapter 8.


Some textbooks of various sorts that may be useful:

Woit, Quantum theory, groups and representations
Carter, Segal and Macdonald, Lectures on Lie groups and Lie algebras
Knapp,  Lie Groups: Beyond an Introduction
Fulton and Harris, Representation theory
Bump, Lie groups
Sepanski, Compact Lie groups
Wallach, Symplectic geometry and Fourier analysis
Kirillov, Lectures on the orbit method

Online Lecture Notes:

The following selection of on-line lecture notes and course materials may be useful:

Lecture notes from Andrei Okounkov's 2016-7 course:  Fall 2016, Spring 2017

Berkeley Lectures on Lie Groups and Quantum Groups

David Ben-Zvi course on representations of SL2, see notes on this page.


Representations of Lie groups and the orbit method, Michele Vergne

Other sources:

Mikhail Khovanov's web page of links.