Perspectives in Mathematical Sciences I

Hiroyuki OCHIAI Professor

Department: Graduate School of Mathematics

Class Time: 2009 Spring Thursday
Recommended for: School of Mathmatics

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Course Overview

This course is designed to be one of the English courses which the Graduate School of Mathematics is providing for graduate and undergraduate students not only from foreign countries but also domestic students who have strong intention to study abroad or to communicate with foreign scientists in English. All course activities - including lectures, homework assignments, questions and consultations- are given in English. The purpose of this course is to introduce and explain various subjects in mathematics. This year, the course is provided by 3 instructors. Each instructor covers different topics from various aspects of mathematics and related fields.

Key Features

This course is one of the English lessons conducted by the mathematical division. Personally, I(Ochiai) am not good at English, but because English is essential in mathematical research, I have, to an extent, experience in English mathematical papers, English lectures, and asking questions (answering them, too) in English. By beginning to use English in mathematics at an early stage in the student's academic education, I hope to lessen the effects of 'English allergy', and hopefully lead the students off to a smooth start in mathematical research. If conditions permit, I would like to make the lessons more interactive, with more emphasis on exercises.

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The Purpose of the Course

We denote by SL(2,R) the set of all regular real matrices of size two. SL(2,R) is a group and a non-compact manifold, that is, one of typical examples of non-compact connected simple Lie groups. We survey the classification of equivalence classes of irreducible unitary representations of SL(2,R), which has been established more than 50 years ago.

The Plan of the Course

This is the first part of three series of lectures. We start from the linear fractional transformation. By differentiating the group action induced on the function spaces, we introduce the simple three-dimensional Lie algebra sl(2) and its representations. The correspondence between Lie groups and Lie algebras are explained in this example. We also compute the basic material on the structure of Lie algebras, such as Killing forms and Casimir operators. We analyze the representations constructed here by using weights and raising/lowering operators. We discuss the irreducibility and the unitarity. As a conclusion, the irreducible unitary representations of SL(2,R) is classified into the following classes: principal series representations (spherical and non-spherical), complementary series representations, (holomorphic and anti-holomorphic) discrete series representations and limit of discrete series representations, and trivial representations.


Lie group, Lie algebra, weight, irreducible, unitary, representation

Required Knowledge

Level 1 is assumed. [An explanation of Level in Department of Mathematics is given in the web page See the precise description in core-2.]


[1] R. Howe and E.C. Tan, Non-abelian Harmonic Analysis, Springer Verlag.
[2] A. Knapp, Representation Theory of Semisimple Groups, Princeton University Press.
[1] is the textbook. A part of [1] will be discussed in the lecture.
[2] is the reference book. The full detail of the story is given in [2].

Course Schedule

date Contents
4/16 correspondence of Lie algebra within the Lie group
4/23 operation of groups and differential representation
4/30 representations of sl(2) and their weights
5/7 unitary, the definition of irreducibility and their classification

*The remaining sessions are conducted by other instructors.


Attendance and reports

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Lecture Handouts

Session #1
Leibniz rule (PDF, 48KB)
Session #2
Linear fractional transformations (PDF, 74KB)
Session #3
Representation (PDF, 84KB)
Session #4
Unitary representation (PDF, 67KB)

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Page last updated June 25, 2009

The class contents were most recently updated on the date indicated. Please be aware that there may be some changes between the most recent year and the current page.

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