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Two commuting symplectomorphisms of a symplectic manifold give rise toactions on Floer cohomologies of each other. We prove the elliptic relationsaying that the supertraces of these two actions are equal.  In the case when a symplectomorphism $f$ commutes with a symplecticinvolution, 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 locusof the involution. We apply this bound to prove that Dehn twists aroundvanishing Lagrangian spheres inside most hypersurfaces in Grassmannians haveinfinite order in the symplectic mapping class group.

Commuting symplectomorphisms and Dehn twists in divisors