  We study the GIT quotient of the symplectic grassmannian parametrizing
lagrangian subspaces of \bigwedge^3{\mathbb C}^6 by the natural action of SL_6,
call it M. This is a compactification of the moduli space of smooth double
EPW-sextics. We determine the stable points, the irreducible components of the
GIT boundary and their dimensions. There are strong analogies with the moduli
space of cubic 4-folds: we prove a result which is analogous to a theorem of
Laza stating that cubic 4-folds with simple singularities are stable. Our final
goal (not achieved in the present paper) is to understand the period map from M
to the Baily-Borel compactification of the relevant period domain modulo an
arithmetic group. With this goal in mind we analyze the locus in the
GIT-boundary of M where the period map is not regular: we prove that it has two
irreducible components, one of dimension 3, the other of dimension 1. Our
results suggests that M is isomorphic to Looijenga's compactification
associated to 3 specific hyperplanes in the period domain.
