Banach-Hecke algebras and p-adic Galois Representations
Abstract
We take some initial steps towards illuminating the (hypothetical)
p
-adic local Langlands functoriality principle relating Galois representations of a
p
-adic field
L
and admissible unitary Banach space representations of
G(L)
when
G
is a split reductive group over
L
. The outline of our work is derived from Breuil's remarkable insights into the nature of the correspondence between 2-dimensional crystalline Galois representations of the Galois group of $\Qdss_p$ and Banach space representations of $GL_{2}(\Qdss_p)$.