  We construct global cohomology classes in the middle degree cohomology of the
Shimura variety of the symplectic group $GSp_6$ compatible when one varies the
level at $p$. These classes are expected constituents of an Euler system for
the Galois representations appearing in these cohomology groups. As an
application, we show how these classes provide elements in the Iwasawa
cohomology of these representations and, by applying Perrin-Riou's machinery,
$p$-adic L-functions associated to them.
