Math 534, Spring 2014

 


Course Title: Lie Theory, II


Instructor: Nigel Higson


Meeting Times: Mondays, Wednesdays and Fridays, 3:30-2:20 in 113 Osmond.


Office Hours:  By appointment, or just try your luck and stop by my office, 228 McAllister.


Prerequisites:  I’ll take the “II” in the course title to mean that I can presume basic familiarity with smooth manifolds, the definition of Lie groups and Lie algebras, the exponential map, Lie subgroups and so on.  See me if you have concerns about your background knowledge (and see also the further remarks below). 


Overview:   The main topic of the course will be representation theory. During the first part of the course we shall study the finite-dimensional representations of compact Lie groups from various points of view. The main objective will be the Weyl character formula. Then we’ll turn to noncompact groups and infinite-dimensional representations.  This is considerably more complicated, and at several places we shall limit ourselves to illustrating theorems in examples rather than proving them in general. Our main objectives will be introductions to the Plancherel formula for complex groups and to the Langlands classification.  We shall also examine discrete series representations.


Texts:  There will be no official textbooks.  There are scores of monographs on Lie theory, both in the library and online, and collectively they cover the material in at least the first part of the course many times over (the second part is less well covered).  I’ll make suggestions as we go along, but here are some references to get started.


Part Zero - Remedial Reading on Lie Groups


If you need to re-acquaint yourself with the basics, you might try the following texts:


    Spivak, Comprehensive Introduction to Differential Geometry, Volume 1, Chapter 10

    Bump, Lie Groups, Sections 5,6,7,8

    Warner, Foundations of Differentiable Manifolds and Lie Groups, Chapters 1,3

    Varadarajan, Lie Groups, Lie Algebras and Their Representations, Chapters 1,2


As for online notes, you might try (the relevant parts of) these:


    Hall - An Elementary Introduction to Groups and Representations

        http://arxiv.org/abs/math-ph/0005032

    Meinrenken - Lie groups and Lie algebras

        http://www.math.toronto.edu/mein/teaching/lie.pdf

    Milicic - Lectures on Lie groups

        http://www.math.utah.edu/~milicic/Eprints/lie.pdf

    Varadarajan - Lie groups

        http://www.math.ucla.edu/~vsv/liegroups2007/liegroups2007.html

    Ziller - Lie groups, representation theory and symmetric spaces

        http://www.math.upenn.edu/~wziller/math650/LieGroupsReps.pdf



Part One - Compact Groups and the Weyl Character Formula


The rudiments of representation theory for compact groups are patterned after the theory for finite groups, and for this there is nothing better than Serre’s book:


    Serre - Linear representations of finite groups, Chapters 1,2


As for the theory for compact Lie groups, here are some texts:


    Brocker and tom Dieck - Representations of compact Lie groups

    Duistermaat, Kolk - Lie groups

    Adams - Lectures on Lie groups


And here are some lecture notes, try these:


     Berkeley notes on Lie groups and Lie algebras

        http://math.berkeley.edu/~anton/written/LieGroups/LieGroups.pdf

    Samelson - Notes on Lie algebras

        http://www.math.cornell.edu/~hatcher/Other/Samelson-LieAlg.pdf

    Sternberg - Lie algebras

        http://www.math.harvard.edu/~shlomo/docs/lie_algebras.pdf

    Varadarajan - Lie groups

        http://www.math.ucla.edu/~vsv/liegroups2007/liegroups2007.html

 

Part Two - Noncompact Groups


The literature on noncompact groups is a bit sparser, but here are some sources:


    Bump - Lie groups

    Howe, Tan - Non-abelian harmonic analysis. Applications of SL(2, R)

    Varadarajan - Harmonic analysis on semisimple Lie groups


And here are two standard tomes, which can be intimidating:

  

    Wallach - Real reductive groups, vol 1

    Knapp - Representation theory of semisimple groups


These discuss the unitary representations of SL(2,R) toward the end:


    Berkeley notes on Lie groups and Lie algebras

        http://math.berkeley.edu/~anton/written/LieGroups/LieGroups.pdf



Homework:  I’ll hand out occasional homework problems to help us deepen our understanding of the material that we’ll encounter: it is important to do calculations here and there (in fact it’s important to do them everywhere).  I’ll ask you to hand in some, but not all of the homework (as we move to more advanced topics I’ll ask you to hand in less and less).


Exams:  There will be none.


Academic Integrity: Students must meet University and the College standards of academic integrity. The University defines academic integrity as "the pursuit of scholarly activity in an open, honest, and responsible manner." It goes on to say that "academic integrity includes a commitment not to engage in or tolerate acts of falsification, misrepresentation or deception. Such acts of dishonesty violate the fundamental ethical principles of the University community and compromise the worth of work completed by others."  See this page.  For a more compelling account of what honesty and integrity should mean, at least for a scientist (or a mathematician), consider these famous words of Richard Feynman. 


Disability Statement: Penn State welcomes students with disabilities into the University's educational programs. If you have a disability-related need for reasonable academic adjustments in this course, contact the Office for Disability Services (ODS) at 814-863-1807 (V/TTY). For further information regarding ODS, please visit the Office for Disability Services Web site at http://equity.psu.edu/ods/.  In order to receive consideration for course accommodations, you must contact ODS and provide documentation (see the documentation guidelines at http://equity.psu.edu/ods/guidelines/documentation-guidelines). If the documentation supports the need for academic adjustments, ODS will provide a letter identifying appropriate academic adjustments. Please share this letter and discuss the adjustments with your instructor as early in the course as possible. You must contact ODS and request academic adjustment letters at the beginning of each semester.



Nigel Higson - 2014