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Abstract: Certain Hilbert spaces, called automorphic representations, are basic
objects in contemporary number theory.
An automorphic representation is a representation of a classical linear
group on a Hilbert space which satisfies a very strong symmetry
condition with respect to the integral points of the group. Though they
generalize Dirichlet characters, automorphic representations are
typically infinite dimensional. However, it may happen that an
automorphic representation contains a canonical vector, called a
newform, which encapsulates the number theoretic information hidden in
the automorphic representation. This is true for automorphic
representations of the symmetry group of the classical upper half
plane. In this talk we will discuss joint work with Ralf Schmidt which
indicates that automorphic representations of the symmetry group of the
Siegel upper half space also admit newforms.
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