This is the second course in our Lie Theory sequence. I shall discuss the structre and representation theory of real Lie groups.
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 | Scan | ||||
| M 8/5 | The principal series |
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