p-adic families and Galois representations for GS_p(4) and GL(2)

In this brief paper, we prove local-global compatibility for holomorphic Siegel modular forms with Iwahori level. In previous work, we proved a weaker version of this result (up to a quadratic twist) and one of the goals of this paper is to remove this quadratic twist by different methods, using p-adic families. We further study the local Galois representation at p for nonregular holomorphic Siegel modular forms. Then we apply the results to the setting of modular forms on GL(2) over a quadratic imaginary field and prove results on the local Galois representation ℓ, as well as crystallinity results at p