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,

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,

Tuesday, April 18:

Thursday, April 20:

Tuesday, April 25: Relation to modular forms

Thursday, April 27:

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.

**Background:**

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.

Textbooks

Some textbooks of various sorts that may be useful:

Woit,

Carter, Segal and Macdonald,

Knapp,

Fulton and Harris,

Bump,

Sepanski,

Wallach,

Kirillov,

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

Mikhail Khovanov's web page of links.