Matthias Strauch. (Cambridge)
p-adic Galois representations and geometric constructions of
Banach space representations
The classical Local Langlands correspondence sets up a
bijection between (isomorphism classes of) n-dimensional representations
of the Weil-Deligne group of a given local field F, and (isomorphism
classes of) irreducible smooth representations of GL(n,F).
This correspondence has the property that the representations on the
Galois side under consideration are trivial on an open subgroup of the
wild inertia group.
In particular, this correspondence neglects almost all Galois
representations which are p-adically continuous (p being the residue
characteristic of F).
In recent years a picture emerged how one may possibly extend this
correspondence to include certain p-adic Galois representations. The
objects on the automorphic side are continuous representations of
GL(n,F) on Banach spaces.
In this talk we will give an overview of the basic ideas behind this
p-adic Langlands correspondence, discuss some results, and explain a
construction of Banach space completions of supercuspidal
representations of GL(2,F) (the latter being joint work in progress with