Project IV (MATH4072) 2018-19


Topics in representation theory

Jens Funke

Description

Based on the representation theory III course there many topics one consider depending on your interest. Here are some suggestions:

1) The Heisenberg group and the Weil representation
  • The Heisenberg group of upper triangular 3x3 matrices and its generalizations are ubiquitous in modern mathematics! Its representation theory has numerous applications in several fields, such as number theory, harmonic analysis, or theoretical physics.
  • You have learned that over the finite field F_p of p elements there is exactly one irreducible representation for each central character. What about if we replace F_p by any Z_n? What are now the irreducible representations etc.?
  • For the Heisenberg group over the real numbers R there is the famous Stone-von Neumann Theorem which states that again for each central character there is one irreducible (unitary) representation. But now this representation is infinite dimensional!
  • With these uniqueness results for both F_p and R one can construct a (projective) representation for SL_2(F_p) and SL_2(R) respectively, which is so important that it runs under multiple names: The Segal-Shale or Weil or Oscillator or Metaplectic or Harmonic Representation. What are its properties?
  • This gives one motivation to study the representation theory of SL_2(F_p) and SL_2(R) (in the latter case, infinite dimensional).  

2) The (Finite-dimensional) Representation Theory of general Lie Groups and Lie Algebras

  • In the course, we covered the situation for SL_2. What about other (so-called) classical groups, such as SL_n, orthogonal, or symplectic groups?

3) Unitary Representations of SL_2(R)

  • We learned that all finite-dimensional representations are non-unitarizable. So for unitary representations we need to look for infinite dimensional ones? What are they? How can you construct them? What are they good for in number theory and physics?

4) Fourier Analysis on Finite Groups and Applications

  • There is wonderful book by A. Terras on this topic which has some very interesting applications.

Resources

There are many books sources on this topic, and I will give an extensive reading list at a later stage.

Prerequisites

  • Representation Theory III

email: J Funke