AFFINE GROUP AS SYMPLECTIC MANIFOLD

For a Lie group carrying a left invariant symplectic form certain Lagrangian foliations are considered. We describe all left invariant (exact) symplectic structures on the affine group K(n) x GL(K(n)) for K=R or C. It is shown that for some of these structures there exists an exact symplectic embeding of the real affine group onto an open subset of the cotangent bundle of the Euclidean group R(n) x SO(n).