Description
The SegalShale 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 Stonevon 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 nondegenerate 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
email: J Funke
