Representations of Algebraic Groups, Autumn 2016

master course, Radboud University Nijmegen

B.J.J. Moonen, Radboud University Nijmegen. Email: B.Moonen at -science.ru.nl-

The lectures are given on Tuesdays from 10:45 to 12:30 in lecture room HG03.054. The course starts in week 36 (September 6, 2016) and continues until week 51 (December 20). There is a break in week 43, and possibly also in week 44. (To be decided.)

The course is intended as an introductory course about representation theory of (mainly reductive) algebraic groups and Lie algebras, or, what amounts to essentially the same, reductive complex Lie groups. We assume a good background in Linear Algebra and in general Algebra (groups, rings, fields). It is useful if you have seen some geometry before, such as the notion of a differentiable manifold. It may also help if you have already seen some notions from representation theory (e.g., in the context of representation theory of finite groups), or if you have already taken a course in Algebraic Geometry or in Lie theory. None of this is a prerequisite, however.

Also on Tuesdays, after this course, Professor Heckman is teaching a course on Coxeter groups. While it is not indispensable to follow this course, it is highly recommended that you do, as you will get to see a far more complete picture.

We mostly follow the book *Representation Theory, a first course* by Fulton and Harris. This book is a gem, and it is highly recommended to buy your own copy. In the course we shall strive to preserve much of the 'hands-on' flavour of the book. For some parts on Lie algebras, the book *Introduction to Lie algebras and Representation Theory* by J. Humphreys is also highly recommended.

Each week we will propose some exercises, It is strongly recommended that you try to do these and hand in (some of) your exercises to Milan Lopuhaä; he will then correct your work and give feedback. You may also discuss the exercises during the Question & Answer sessions, which are on Tuesdays from 16:45 to 17:30.

During the semester there will be three hand-in assignments. To pass the course, you should have a sufficient average grade (at least 6) for the hand-in assignments and then finish by doing an oral exam. For these oral exams, the following time slots are available, which are assigned on a first come, first served basis. You are requested to send me an email with your preferred day and time; do give me one or two alternatives. If you are unable to take the oral exam on 26 or 27 January, let me know. Here is a description of what you should know for the oral exam.

Day and time | Student |
---|---|

Thurs, Jan 26, 10:00-10:45 | |

Thurs, Jan 26, 10:45-11:30 | Paulien Schets |

Thurs, Jan 26, 11:30-12:15 | Lieke-Rosa Koetsier |

Thurs, Jan 26, 13:00-13:45 | Steve Alberts |

Thurs, Jan 26, 13:45-14:30 | Jeroen Hekking |

Fri, Jan 27, 9:00-9:45 | Jeroen Winkel |

Fri, Jan 27, 9:45-10:30 | |

Fri, Jan 27, 10:30-11:15 | Krijn Reijnders |

Fri, Jan 27, 11:15-12:00 | |

Fri, Jan 27, 12:00-12:45 |

Date | Topics |
---|---|

Sept 6 | Algebraic groups. Orthogonal and symplectic groups. Notes. Exercises. (Note that the last part of Exercise 2(b) was corrected; the original version contained a mistake.) |

Sept 13 | Multilinear algebra; symmetric and exterior powers. (Lecture by Erik Koelink.) Notes. Exercises. |

Sept 20 | Algebraic homomorphisms. Basic notions from representation theory. Notes. Exercises. |

Sept 27 | Representations of tori. Irreducible and semisimple representations. Isotypic components. Notes. Here is the first hand-in assignment, which is due the 11th of October. As always, what you hand in should be your own work. |

Oct 4 | The Lie algebra of an algebraic group. Notes. Exercises. |

Oct 11 | Representations of SL_{2} and sl_{2}. You should now also start reading Fulton-Harris, especially Chapter 11. Notes. Exercises. |

Oct 18 | Solvable and nilpotent Lie algebras. Reading material: the Fulton-Harris book, sections 9.1 and 9.2, and/or the book by Humphreys, Chapter 3. Exercises. |

Oct 25 | break |

Nov 1 | Semisimple Lie algebras. Notes. Reading material: Fulton-Harris, Appendix C, Humphreys, Sections 5 and 6. Exercises. |

Nov 8 | Representations of sl_{3} (first part). Notes (incomplete). Reading: Fulton-Harris, Chapters 12 and 13. Exercises. |

Nov 15 | Representations of sl_{3} (part 2). Reading: Fulton-Harris, Chapters 12 and 13. Here is the second hand-in assignment, which is due the 29th of November. As always, what you hand in should be your own work. |

Nov 22 | The root system associated with a semisimple Lie algebra. Notes. (This is more than what I have managed to cover.) Exercises. Exercise 4 involves things we have not yet discussed in class. |

Nov 29 | Root systems and weights. Notes. Exercises. |

Dec 6 | Root systems and weights - continuation and examples. Exercises. |

Dec 13 | Representations of semisimple groups. Here is the third and final hand-in assignment, which is due the 9th of January. |

Dec 20 | The Weyl Character Formula and Freudenthal's Multiplicity Formula |

To the webpage of Ben Moonen.