Banach-Hecke algebras and p-adic Galois Representations

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 ${\text{Qdss}}_p$ and Banach space representations of $G{L}_2({\text{Qdss}}_p)$.