## MATH 534, Spring 2020

Course Title: Lie Theory, II

Instructor: Nigel Higson

Meeting Times: Tuesdays and Thursdays, 10:35-11:50 in 204 Sackett.

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

Prerequisites: You should have 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 the representation theory of Lie groups. During the first part of the course we shall study the finite-dimensional representations of compact Lie groups from various points of view (Hilbert space theory, complex geometry, Lie algebra theory). The main objectives will be the classification of irreducible representations and the Weyl character formula. Then we shall turn to noncompact groups and infinite-dimensional representation theory, which is a more complicated topic. We shall examine (but certainly not exhaust) Harish-Chandra’s theory of tempered unitary representations and the Langlands classification.

Texts: There will be no official textbooks, but here is some recommended reading.

The second part is less well covered, shall try to provide at least some lecture notes.

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

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. 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.

Instructor: Nigel Higson

Meeting Times: Tuesdays and Thursdays, 10:35-11:50 in 204 Sackett.

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

Prerequisites: You should have 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 the representation theory of Lie groups. During the first part of the course we shall study the finite-dimensional representations of compact Lie groups from various points of view (Hilbert space theory, complex geometry, Lie algebra theory). The main objectives will be the classification of irreducible representations and the Weyl character formula. Then we shall turn to noncompact groups and infinite-dimensional representation theory, which is a more complicated topic. We shall examine (but certainly not exhaust) Harish-Chandra’s theory of tempered unitary representations and the Langlands classification.

Texts: There will be no official textbooks, but here is some recommended reading.

The second part is less well covered, shall try to provide at least some lecture notes.

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

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. 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.