Gerrit Van Dijk, “GELFAND PAIRS AND BEYOND,” COE Lecture Note Series 11, 2008

Between April and May 2008, an intensive lecture series was conducted by Professor Gerrit Van Dijk. Based on the content of these lectures, the “COE Lecture Note Series 11,” titled GELFAND PAIRS AND BEYOND, was published. This volume provides a comprehensive exploration of key topics in representation theory, ranging from finite groups to infinite-dimensional representations and generalized Gelfand pairs.

Outline:

1. Representations and characters of finite groups
2. Representations of compact groups
3. Characters of infinite-dimensional representations
4. Representations of compact groups (cont.)
5. Spherical harmonics
6. Compact Glefand pairs
7. Gelfand pairs
8. Generalized Gelfand pairs

A Notable Exercise

One of the exercises in this lecture series posed the following question:

Prove that \( (SU(2), SO(2)) \) is a compact Gelfand pair.

To solve this problem, I utilized the maximal torus \( T \) of \( SU(2) \) and decomposed \( SU(2) \) as: \[ SU(2) = SO(2) T SO(2). \] This approach turned out to be successful, and Professor Van Dijk praised it, saying, “Good idea!” This solution was later referenced in the lecture notes, where my name was credited.

Details of the Decomposition and an Alternative Approach

While my solution involved decomposing \( SU(2) \) by skillfully adjusting angles, Professor Van Dijk later explained an alternative method. Specifically:

• He later explained that since \( S^3 \) is a homogeneous space of \( SU(2) \), we can find a fixed subgroup of \( \mathbf{e}^1 \) and decompose it.

This method is widely used in cases where Lie groups act transitively on a space and is often associated with well-known decomposition techniques. Employing such approaches effectively reveals the structure of the group and facilitates the clear presentation of theoretical concepts.

Insights Gained from the Lecture

Through this lecture series, I not only deepened my understanding of theoretical concepts but also learned valuable problem-solving strategies and the importance of approaching problems from diverse perspectives. The theme of Gelfand pairs, in particular, holds profound significance in harmonic analysis and the theory of Lie groups, making this lecture series an invaluable learning experience.

The book GELFAND PAIRS AND BEYOND is highly recommended for researchers interested in representation theory and harmonic analysis. It serves as an excellent reference for exploring these advanced topics.