Project IV (MATH4072) 2024-25

The finite Weil representaton

Jens Funke

Description

The Segal-Shale or Weil or oscillator or metaplectic or harmonic representation comes with many names corresponding to its ubiquitous nature in pure mathematics but also mathematical physics.

In this project, which should be taken in parallel with the representation theory IV course, we look at one particular variant, the finite Weil representation over a finite field F_p (or more generally the ring Z/nZ), motivated by applications in number theory and modular forms. Here are some aspects
  • 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. We can then consider its representation theory which is remarkably simple, namely the famous Stone-von Neumann Theorem and its analogues.
  • One can then use this to explicitly construct the finite Weil representation, which for an odd prime p is a representation of the special linear group SL_2(F_p).
  • The finite Weil representation can be constructed more generally for any finite dimensional vector space V over a finite field equipped with a non-degenerate form, and as such it becomes also a representation of the finite orthogonal group O(V) preserving the underlying bilinear form.
  • The actions of SL_2(F_p) and O(V) actually commute, and with this one obtains a correspondence between representations of these two groups which is known as the theta respectively the Howe correspondence. We will study this correspondence, putting particular emphasis on the case of the trivial representation (which is not trivial at all!).
  • In the process we will also study the representation theory of SL_2(F_p) and of finite orthogonal groups which nicely complements the RT IV module.
  • In case of interest one can also study the situation over the real numbers R.

Resources

There are a good number of research articles and other sources on this topic, to be provided at a later stage.

Corerequisites

  • Representation Theory IV

email: J Funke