  Two commuting symplectomorphisms of a symplectic manifold give rise to
actions on Floer cohomologies of each other. We prove the elliptic relation
saying that the supertraces of these two actions are equal.
  In the case when a symplectomorphism $f$ commutes with a symplectic
involution, the elliptic relation provides a lower bound on the dimension of
$HF^*(f)$ in terms of the Lefschetz number of $f$ restricted to the fixed locus
of the involution. We apply this bound to prove that Dehn twists around
vanishing Lagrangian spheres inside most hypersurfaces in Grassmannians have
infinite order in the symplectic mapping class group.
