## MATH 534, Spring 2022

Our main objectives for compact groups will be Weyl’s character formula, Verma modules and the Borel-Weil theorem. As for noncompact groups, the work of Harish-Chandra on tempered representations is a towering achievement in analysis, and we shall not do much more than sketch the main themes. The Langlands classification is a bridge from Harish-Chandra’s theory to algebra, algebraic geometry, and beyond. If all goes according to plan, then we shall be able to discuss Langlands' proof in some detail. Throughout, there will be an emphasis on examples that illustrate well the general case (mostly SU(3) for compact groups and SL(3,R) for noncompact groups), rather than a full treatment of the most general case.

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

Or you might try (the beginning parts of) these online notes:

There are scores of monographs on Lie theory, both in the library and online, and collectively they cover the material in the first part of the course many times over, and very well. But you might begin with the Segal’s really wonderful contribution to this volume:

We shall follow Segal’s approach quite closely when we examine the link between representations and complex analytic functions.

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:

Here are some good books that focus more specifically on representation theory for compact groups:

And here are some online lecture notes to try, in addition to the ones I’ve already listed:

The latter two focus on the Lie-algebraic approach to the theory, and we’ll follow Sternberg’s notes when we cover that topic in class.

The literature on noncompact groups is much sparser. Here are two sources that treat aspects of SL(2,R):

In addition, the Berkeley notes listed above discuss the unitary representations of SL(2,R) toward the end, as does Segal’s paper. Varadarajan’s book has a great deal to say about the Harish-Chandra theory, and we shall rely on it here and there.

For the record here are the two standard tomes, which can be intimidating:

Of the two, I recommend Wallach's book the most, but be prepared for a struggle if you pick it up.

**Course Title:**Lie Theory, II**Instructor:**Nigel Higson**Meeting Times:**Tuesdays and Thursdays, 10:35-11:50, via Zoom, at least initially.**Office Hours:**By appointment. Send me an email.**Prerequisites:**You should have some basic familiarity with 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 class will concentrate on representations of Lie groups. Here is what we shall cover:- Representation theory for compact groups from the perspectives of analysis, algebra and geometry.
- Parabolic induction for real reductive groups and discrete series representations
- An introduction to Harish-Chandra’s theory of tempered unitary representations.
- An introduction to the Langlands classification.

Our main objectives for compact groups will be Weyl’s character formula, Verma modules and the Borel-Weil theorem. As for noncompact groups, the work of Harish-Chandra on tempered representations is a towering achievement in analysis, and we shall not do much more than sketch the main themes. The Langlands classification is a bridge from Harish-Chandra’s theory to algebra, algebraic geometry, and beyond. If all goes according to plan, then we shall be able to discuss Langlands' proof in some detail. Throughout, there will be an emphasis on examples that illustrate well the general case (mostly SU(3) for compact groups and SL(3,R) for noncompact groups), rather than a full treatment of the most general case.

**Texts:**There will be no official textbook, but here is some recommended reading.*Background 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

Or you might try (the beginning parts of) these online notes:

- 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*There are scores of monographs on Lie theory, both in the library and online, and collectively they cover the material in the first part of the course many times over, and very well. But you might begin with the Segal’s really wonderful contribution to this volume:

- Carter, Segal, Macdonald - Lectures on Lie groups and Lie algebras

We shall follow Segal’s approach quite closely when we examine the link between representations and complex analytic functions.

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

Here are some good books that focus more specifically on representation theory for compact groups:

- Brocker and tom Dieck - Representations of compact Lie groups
- Duistermaat, Kolk - Lie groups
- Adams - Lectures on Lie groups

And here are some online lecture notes to try, in addition to the ones I’ve already listed:

- 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

The latter two focus on the Lie-algebraic approach to the theory, and we’ll follow Sternberg’s notes when we cover that topic in class.

*Part Two - Noncompact Groups*The literature on noncompact groups is much sparser. Here are two sources that treat aspects of SL(2,R):

- Howe, Tan - Non-abelian harmonic analysis. Applications of SL(2, R)
- Varadarajan - Harmonic analysis on semisimple Lie groups

In addition, the Berkeley notes listed above discuss the unitary representations of SL(2,R) toward the end, as does Segal’s paper. Varadarajan’s book has a great deal to say about the Harish-Chandra theory, and we shall rely on it here and there.

For the record here are the two standard tomes, which can be intimidating:

- Wallach - Real reductive groups, vol 1
- Knapp - Representation theory of semisimple groups

Of the two, I recommend Wallach's book the most, but be prepared for a struggle if you pick it up.

**Lecture Notes:**I'll provide you with access to the typed lecture notes from the previous version of this course, and during the semester I shall edit those notes to match this year's lectures as closely as possible.**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 no exams.**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.