Based on the representation theory III course there many
topics one consider depending on your interest. Here are
1) The Heisenberg group and the
- 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
- 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, you learned about the
finite-dimensional representation theory of SL_2. What
about other (so-called) classical groups, such as SL_n,
orthogonal, or symplectic groups?
3) Unitary Representations of
- You 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?
There are many books sources on this topic, and I will give
an extensive reading list at a later stage.
- Representation Theory III
email: J Funke