Math 636. Representation Theory (Spring 1999)

Title: Representation Theory
Meeting times: MWF.......9:00am-9:50am (MTH 0302)
Instructor: Professor Jonathan Rosenberg. His office is room 2114 of the Math Building, phone extension 55166, or you can contact him by email. His office hours are Mondays 10-11 and 2-3, or by appointment.
Text: Noncommutative Harmonic Analysis, by Michael E. Taylor, Mathematical Surveys and Monographs, Volume 22, American Mathematical Society, 1986 (reprinted 1990), ISBN: 0-8218-1523-7.
Prerequisite: MATH 630-631, plus elementary abstract algebra (at the level of MATH 402 or 403). A little elementary complex analysis (at the level of MATH 463) will also be useful.
Catalog description: Introduction to representation theory of Lie groups and Lie algebras; initiation into non-abelian harmonic analysis through a detailed study of the most basic examples, such as unitary and orthogonal groups, the Heisenberg group, Euclidean motion groups, the special linear group. Additional topics from the theory of nilpotent Lie groups, semisimple Lie groups, p-adic groups or C*-algebras.

Course Description:

After a quick discussion of some topics from classical harmonic analysis, viewed from the perspective of the representation theory of locally compact abelian groups, the course will focus on some important examples of what representation theory of Lie groups is all about. The emphasis will be on a wide variety of applications: to algebra, to partial differential equations, to mathematical physics, to geometry, and to number theory. The course will not be a detailed study of the structure theory of Lie groups, which will be covered next year in MATH 744. The examples treated will therefore be those that can be understood by a general mathematical audience without knowing such structure theory, particularly, SU(2) and SO(3) as prototypes of noncommutative compact Lie groups, the Heisenberg group and ax + b groups as prototypes of noncommutative solvable Lie groups, and SL(2, R) as a prototype of semisimple Lie groups.

Course Outline

  1. Basics of representations (Schur's Lemma, etc.) (Taylor, Ch. 0)
  2. Characters and harmonic analysis on Zn, Tn, and Rn. (Taylor, Appendix A)
  3. Some applications of commutative harmonic analysis (PDEs, etc.)
    There is a homework assignment on this material, due Feb. 15.
  4. Compact and trace-class operators, trace formulas
    There is a homework assignment on this material, due Mar. 1.
  5. Basics of character theory of compact groups, Peter-Weyl Theorem (Taylor, Ch. 3)
    Some notes are available.
  6. Representation theory of the symmetric groups Sn and of the compact Lie groups SO(3) and SU(2) (Taylor, Ch. 2)
    There is a homework assignment on this material, due Mar. 29.
  7. Applications of representations of SO(3) and SU(2) to differential geometry and physics (Taylor, Ch. 2 and 4)
  8. Induced representations and the Imprimitivity Theorem (Taylor, Ch. 5)
    There is a homework assignment on this material, due Apr. 26.
  9. Representation theory of the Heisenberg group (Taylor, Ch. 1)
  10. Analysis on Heisenberg nilmanifolds. Applications to differential geometry, PDEs, number theory and algebraic geometry
    There is a homework assignment on this material, due May 7.
  11. Representation theory of SL(2, R) (Taylor, Ch. 8)
  12. Applications of representations of SL(2, R) to number theory, geometry, complex analysis (Taylor, Ch. 8)