Banach-Hecke algebras and p-adic Galois Representations

TL;DR

This paper explores the relationship between p-adic Galois representations and Banach space representations of reductive groups over p-adic fields, aiming to advance the p-adic local Langlands program.

Contribution

It initiates the study of a hypothetical p-adic local Langlands functoriality linking Galois representations and Banach space representations for split reductive groups over p-adic fields.

Findings

Insights into the structure of Banach-Hecke algebras

Connections between crystalline Galois representations and Banach representations

Foundational steps towards a p-adic local Langlands functoriality

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 $GL_{2}(\Qdss_p)$.