## Banach-Hecke algebras and $p$-adic Galois representations

### Summary

In this paper, 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 $L$.

### Mathematics Subject Classification

11F80, 11S37, 22E50

### Keywords/Phrases

$p$-adic local Langlands correspondence, Satake isomorphism, Iwahori-Hecke algebra, Banach-Hecke algebra, filtered isocrystal, crystalline Galois representation