An involution of a manifold is a self map which is its own inverse (for example a reflection). An antisymplectic involution of a symplectic manifold is an involution which takes the symplectic form to its negative. As a result the real locus (the fixed point set of an antisymplectic involution) is Lagrangian (in other words the symplectic form restricts to zero on the real locus). In a fundamental 1983 paper, Duistermaat proved results relating the topology of the fixed point set and that of the symplectic manifold, where the degree of a generator of the cohomology of the manifold is twice the degree of a generator of the cohomology of the fixed point set. A prototype is the 2-sphere with the involution being reflection in the equatorial plane. The fixed point set is the equator, a unit circle.

If in addition there is a Hamiltonian torus action on the symplectic manifold acting compatibly with the involution, Duistermaat proved that the image of the fixed point set of the involution under the moment map is the same as the image of the entire manifold under the moment map. These objects were studied by Hausmann, Holm and Puppe, who called them conjugation spaces.

In joint work with Liviu Mare, we have showed that the based loop group has an antisymplectic involution for which the torus action is compactible. We have studied the cohomological consequences.