Representation Theory of SL_2 M390C.
Spring Semester 2005. Tue-Th 11-12:30
Office Hours: W 3-4
Reserve: The books by Carter et al. and Varadarajan
are on 3-day loan reserve at the math physics library.
Brief description: I will present an unorthodox
introduction to the representation theory of Lie groups and Lie
algebras, focussing entirely on the group of two by two matrices with
determinant one. Thus we hope to cover a breadth of topics in
representation theory, that are usually sacrificed for the depth of
treating general Lie groups. We will start with finite dimensional
representations of SL_2(C) (or equivalently SU_2) and their relation
to the geometry of the Riemann sphere. We will then move on to unitary
(infinite dimensional) representations of SL_2(C) and especially
SL_2(R), and their relation to the geometry of the upper half plane
and modular forms. We will (in an ideal universe)
conclude with the beautiful geometry of the
tree, associated to SL_2(Q_p), which is a p-adic analog of the upper
Besides its intrinsic beauty, this subject is ubiquitous in number
theory, topology and physics. In final projects we will explore some
of these connections as well as generalizations to other Lie groups.
Basic Lie Theory:
- R. Carter, G. Segal and I.MacDonald: Lectures on Lie Groups
and Lie Algebras. London Math. Society Student Texts 32. The lectures
by Segal are a beautiful overview of the fundamental ideas of Lie
groups and algebras, with geometry and examples emphasized.
- W. Fulton and J. Harris: Representation Theory. Springer GTM 129.
The canonical reference for representations, especially for the (Lie)
algebraic point of view and basic algebro--geometric aspects.
- G. Mackey: Harmonic Analysis as the Exploitation of Symmetry.
Bull. Amer. Math. Soc. 3/1 (1980) 543-699. Reprinted in: The Scope and
History of Commutative and Noncommutative Harmonic Analysis. History
of Math Vol.5, Amer. Math Soc./London Math Soc. 1992. My favorite (and
first) introduction to representation theory, emphasizing its history
and origins in probability, number theory and physics. The reprint
appears in a volume devoted to Mackey's wonderful representation
theory survey articles.
- Other Mackey surveys (besides those in the book above): Unitary
Group Representations in physics, probability and number theory
(Addison-Wesley 1989) -- a book delving deeper into the ideas of the
above survey article.
- W. Schmid: Representations of semi-simple Lie groups. In: Atiyah
et al., Representation Theory of Lie Groups. Londom Math Society
Lecture Notes 34. Cambridge U. Press 1979. Excellent overview by a master,
in a volume full of useful reviews (also Mackey, Bott, Kostant, Kazhdan..)
- W. Schmid: Analytic and Geometric Realization of Representations. In:
Tirao and Wallach (eds)., New Developments in Lie Theory and their Applications.
Lecture notes overviewing the subject in the title, with a strong emphasis on SL_2.
More advanced, SL_2 specific references:
- K. Vilonen: Representations of SL_2. Course Notes, Northwestern
- R. Howe and E.C. Tan: Non-abelian harmonic analysis - applications
of SL(2,R). Springer Universitext. A very nice introduction to
representations of SL(2,R), with interesting applications to classical
- V.S. Varadarajan: An Introduction to Harmonic Analysis on
Semisimple Lie Groups. Cambridge Studies in advanced math 16. Nice
textbook, with an emphasis on SL_2 and good introductions to various
- S. Lang: SL(2,R). Springer GTM. Fairly analytical introduction,
alas not very coherent -- read the review
- D. Ramakrishnan and R. Valenza: Fourier Analysis on Number Fields.
Springer GTM 186. Contains an introduction to the Pontrjagin duality
theory for locally compact abelian groups, and its applications to
number theory (Tate's thesis).
- J. Arthur: Harmonic Analysis and Group Representations. Notices
of the AMS. Gorgeous introduction to the ideas of Harish-Chandra.
Available off AMS website.
- I. Gelfand, M. Graev and I. Piatetskii-Shapiro: Representation Theory
and Automorphic Functions. Academic Press. A classic thorough study
of representations of SL2 over real, p-adic and adelic fields.
- D. Bump: Automorphic Forms and Representations. Cambridge Studs. Advanced
Math. 55. A good and detailed
introduction to Langlands program ideas focussed on representations of GL2.
- T. Bailey and A. Knapp (eds): Representation Theory and Automorphic Forms.
Proc. Symp. Pure Math 61. Proceedings of an instructional conference,
with a variety of great introductory articles. (In particular Schmid, Knapp
- J. Bernstein and S. Gelbart (eds): An Introduction to the Langlands
Program. Birkhauser. A very timely collection of introductions to the
basic constituents of the Langlands program.
- A. Borel and W. Casselman (eds): Automorphic Forms, Representations
and L-functions (the Corvallis volumes). The standard source of information
about the Langlands program. Available off the AMS website
(see e.g. Arinkin's webpage, below).
- J. Bernstein, Courses on Eisenstein Series and Representations
of p-adic Groups. Beautiful expositions. Available at
- Wee Teck Gan,
Automorphic Forms and Automorphic Representations: slides for a series
of five lectures given in Hangzhou, China giving an excellent overview
of the basic theory (available on his web page).