  Let X be a compact manifold with a smooth action of a compact connected Lie
group G. Let $L\to X$ be a complex line bundle. Using the Cartan complex for
equivariant cohomology, we give a new proof of a theorem of Hattori and Yoshida
which says that the action of G lifts to L if and only if the first Chern class
$c\sb 1(L)$ of L can be lifted to an integral equivariant cohomology class in
$H\sp 2\sb G(X;\ZZ)$, and that the different lifts of the action are classified
by the lifts of $c\sb 1(L)$ to $H\sp 2\sb G(X;\ZZ)$. As a corollary of our
method of proof, we prove that, if the action is Hamiltonian and $\nabla$ is a
connection on L which is unitary for some metric on L and whose curvature is
G-invariant, then there is a lift of the action to a certain power $L\sp d$
(where d is independent of L) which leaves fixed the induced metric on $L^d$
and the connection $\nabla\sp{\otimes d}$. This generalises to symplectic
geometry a well known result in Geometric Invariant Theory.
