We consider Lagrangian Floer cohomology for a pair of Lagrangian submanifolds
in a symplectic manifold M. Suppose that M carries a symplectic involution,
which preserves both submanifolds. Under various topological hypotheses, we
prove a localization theorem for Floer cohomology, which implies a Smith-type
inequality for the Floer cohomology groups in M and its fixed point set. Two
applications to symplectic Khovanov cohomology are included.

We construct open symplectic manifolds which are convex at infinity
("Liouville manifolds") and which are diffeomorphic, but not symplectically
isomorphic, to cotangent bundles T^*S^{n+1}, for any n+1 \geq 3.

These manifolds are constructed as total spaces of Lefschetz fibrations,
where the fibre and all but one of the vanishing cycles are fixed. We show that
almost any choice of the last vanishing cycle leads to a nonstandard symplectic
structure (those choices which yield standard T^*S^{n+1} can be exactly
determined).