Speaker: David Rottensteiner

Title: Foundations of Harmonic Analysis on the Heisenberg Group

Abstract: The Heisenberg group $\H^n$ is the "simplest" non-commutative
Lie group and plays an important role in several branches of
mathematics. After constructing the group and its Lie algebra we focus
on the representations of $\H^n$. We will prove a famous theorem by
Marshall Harvey Stone and John von Neumann, classifying the irreducible
unitary representations of the Heisenberg group. Along the way we
present concepts like the twisted convolution and integrated
representations. We finally introduce the group Fourier transform for
$\H^n$ and prove the Plancherel theorem, which establishes an isometric
isomorphism between $L2(\H^n)$ and the space $L2(\R^*,HS(L2(\R^n));
\mu)$ of square-integrable functions from $\R^*$ into the space of
Hilbert-Schmidt operators on $L2(\R^n)$. Here, $\mu$ is the so-called
Plancherel measure.