Winter Term 2023

Lior Silberman
- Office: MATX 1112, 604-827-3031
- Email: "lior" (at) Math.UBC.CA (please include the course number in the subject line, if applicable)
- Office hours (Fall 2024): after class (W 10:00-10:30 at the ESB atrium, W 14:00-14:30 at WW IRC Atrium), M 10:30-11:30 in my office, and by appointment.

- Classes: MWF 14:00-15:00 at ESB 4127 and on Zoom.
- Syllabus.
- (Rough) lecture notes (updated 12/3/2023).

This is the second course in our Lie Theory sequence. I shall discuss the structre and representation theory of real Lie groups.

- BrÃ¶cker–tom Dieck, Representations of Compact Lie Groups
- Knapp, Lie Groups Beyond an Introduction

*Warning: the following information is tentative and subject to change at any time*

Chapter | Week | Date | Material | In-class | Notes |
---|---|---|---|---|---|

Compact topological groups |
1 | M 9/1 | Introduction | Scan | |

W 11/1 | Topological groups; representations | Scan | |||

F 13/1 | Basic constructions | Scan | |||

2 | M 16/1 | Compact groups | Scan | ||

W 18/1 | G-finite vectors | Scan | |||

F 20/1 | The Peter--Weyl Theorem | Scan | |||

Lie groups | 3 | M 23/1 | Manifolds | Scan | |

W 25/1 | Tangent and Cotangent spaces | Scan | |||

F 27/1 | Lie Groups | Scan | |||

4 | M 30/1 | Lie algebras | Scan | ||

W 1/2 | The exponential map | Scan | |||

F 3/2 | Closed subgroups | Scan | |||

5 | M 6/2 | The adjoint representation | Scan | ||

Compact Lie groups |
W 8/2 | Compact Lie groups; tori | Scan | ||

F 10/2 | Centralizers of tori | Scan | |||

6 | M 13/2 | Maximal tori | Scan | ||

W 15/2 | SU(2); weights | Scan | |||

F 17/2 | Roots | Scan | |||

20/2–25/2 | Winter break | ||||

7 | M 27/2 | Groups of rank 1 | Scan | ||

W 1/3 | The algebraic Weyl group | Scan | |||

F 3/3 | Weyl Chambers | Scan | |||

8 | M 6/3 | Root Systems | Scan | ||

W 8/3 | The dual Weyl chamber | Scan | |||

Representation theory of compact Lie groups |
F 10/3 | Represetation theory of SU(2) | Scan | ||

9 | M 13/3 | (continued) | Scan | ||

W 15/3 | The universal enveloping algebra | Scan | |||

F 17/3 | Highest weight thm: uniqueness | Scan | |||

10 | M 20/3 | (continued) | Scan | ||

W 22/3 | Verma modules | Scan | |||

F 24/3 | Weyl itegration formula | Scan | |||

11 | M 27/3 | Weyl character formula | Scan | ||

Semisimple Lie groups |
W 29/3 | Semisimple Lie algebras | Scan | ||

F 31/3 | Cartan involutions | Scan | |||

12 | M 3/4 | Global Cartan involution | Scan | ||

W 5/4 | No lecture due to Passover | ||||

W 12/4 | Symmetric spaces | Scan | |||

13 | M 24/4 | Iwasawa decomposition | Scan | ||

F 28/4 | (continued) | Scan | |||

Infinite dimensional representations |
14 | M 1/5 | Smooth vectors | Scan | |

F 5/5 | ${\text{SL}}_{2}\left(\mathbb{R}\right)$ | Scan | |||

M 8/5 | The principal series |

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