MATH 630

MATH 636

REPRESENTATION THEORY

SPRING 2004

Prof. Herb

Office: 2105 Math Bldg.

Phone: (301) 405-5159

E-Mail: rah@math.umd.edu

Office Hours: Monday 9-10, Wednesday 1-2, and Friday 11-12

Text: None required. I will use material from a variety of textbooks including:
Lie Groups, An Introduction Through Linear Groups, by Wulf Rossmann
Lie Groups, Lie Algebras, and Representations, An Elementary Introduction, by Brian Hall (Springer Graduate Texts 222)
Introduction to Lie Algebras and Representation Theory, by James Humphreys (Springer Graduate Texts 9)
Unitary Representations and Harmonic Analysis, An Introduction, by Mitsuo Sugiura

I will spend the first part of the semester discussing finite dimensional representations, and then move on to
some examples of infinite dimensional unitary representations.

Grading: Grades will be based on homework assignments and possibly class presentations.

For the pdf file of notes on representation theory and harmonic analysis click here: Notes.

Material Covered and Homework Assigned :

M 1/26 - Snow day.
W 1/28 - I gave examples of matrix groups and finite-dimensional representations of them.
F 1/30 - I talked about unitary representations, invariant subspaces, and complete reducibility of finite-dimensional unitary representations.
M 2/2 - I discussed Haar measure, complete reducibility of representations of finite and compact groups, contragredient, and tensor product representation. I will hand out a set of homework problems on Wednesday.
W 2/4 I started 6.2 of Rossman and discussed Schur's Lemma and abelian groups. I handed out the homework assignment which is due Monday, Feb. 16. I announced that class this Friday is cancelled.
F 2/6 class cancelled
M 2/9 - I talked about L^2(G) and matrix coefficients of representations. We are starting a series of results which will lead to the Peter-Weyl theorem for compact groups
W 2/11 - I continued with a proof of the Peter-Weyl theorem.
F 2/13 - I finished the proof. I will go over one more related point, and then start on characters on Monday. The first homework set is due Monday. Today I handed out a second homework set which will be due Monday, Feb. 23.
M 2/16 - I started character theory.
W 2/18 - I continued with character theory.
F 2/20 - We spent most of the hour going over HW 1.
M 2/23 - I continued with character theory and handed out a third set of homework problems, due Monday, March 1.
W 2/25 - I finished character theory and started a discussion of U(n). I announced that the third HW set is not due until Monday, March 8.
F 2/27 - We spent most of the hour going over HW 2.
M 3/1 - I continued with U(n).
W 3/3 - I continued talking about characters of U(n). Note HW #3 is due Monday, March 8.
F 3/5 - I started the proof of the Weyl character formula for U(n).
M 3/8 - I finished the Weyl character formula. Next time I will prove the Weyl dimension formula and start talking about Lie algebras. I handed out HW 4. It is due Monday, March 29.
W 3/10 - I started basic definitions and facts about Lie algebras.
F 3/12 - We went over HW 3 and I continued with facts about Lie algebras.
M 3/15 - I finished preliminaries on Lie algebras and started section 6.5 of Rossmann on representations of Lie algebras
W 3/17 - I talked mostly about the Lie algebra sl(2,R) and its action on polynomials in 2 variables.
F 3/19 - I continued with section 6.5 of Rossmann.
Happy Spring Break
M 3/29 - I finished section 6.5 of Rossmann and handed out HW 5. It is due in 2 weeks (Monday, April 12)
W 3/31 - I talked about the adjoint representation, and SU(2) and SO(3).
F 4/2 - We spent the whole period going over HW 4. Note: There is an error in HW 5, problem 3. (a) is OK, (b) is OK if you cross out the word holomorphic, (c) is OK, (d) is false, even for n=1. I am still trying to figure out what the idea was supposed to be for this part.
M 4/5 - I handed out a revised version of HW 5. I started talking about the Borel-Weil theorem.
W 4/7 - I started the proof of the Borel-Weil theorem.
F 4/9 - I finished the proof of the Borel-Weil theorem except for one important lemma. HW 5 is due Wed. 4/14.
M 4/12 - No class today.
W 4/14 - I finished the Borel-Weil theorem.
F 4/15 - I started talking about reductive complex groups. I handed out HW 6. It is due Monday 4/26.
M 4/19 - We went over problems from HW 5.
W 4/21 - I continued talking about reductive groups.
F 4/23 - I finished the section on reductive groups. Monday HW 6 is due and I will start lectures on infinite-dimensional unitary representations and harmonic analysis.
M 4/26 - I started lectures on infinite-dimensional unitary representations and handed out notes. (You can download a copy of the notes using the above link.
W 4/28 - I talked about locally compact abelian groups and characters of infinite-dimensional representations. I handed out the last homework set, HW 7, due Monday, May 10.
F 4/30 - We went over HW 6 and I started talking about unitary representations of SL(2,C).
M 5/3 - I talked about principal series of SL(2,C) and SL(2,R).
W 5/5 - I talked about the unitary dual of SL(2,R) and discrete series representations.
F 5/7 - We had 4 student presentations on their research involving representation theory.
M 5/10 - I talked about tempered representations.

File translated from T_EX by T_TH, version 2.21.

On 31 Jan 2000, 09:05.