We construct finite group actions on Lagrangian Floer theory when symplectic manifolds have finite group actions and Lagrangian submanifolds have induced group actions. We first define finite group actions on Novikov-Morse theory. We introduce the notion of a spin profile as an obstruction class of extending the group action on Lagrangian submanifold to the one on its spin structure, which is a group cohomology class in H 2 (G; Z/2). For a class of Lagrangian submanifolds which have the same

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... n profiles, we define a finite group action on their Fukaya category. In consequence, we obtain the s-equivariant Fukaya category as well as the s-orbifolded Fukaya category for each group cohomology class s. We also develop a version with G-equivariant bundles on Lagrangian submanifolds, and explain how character group of G acts on the theory. As an application, we define an orbifolded Fukaya-Seidel category of a G-invariant Lefschetz fibration, and also discuss homological mirror symmetry conjectures with group actions. C.-H. Cho and H. Hong 8 Group actions on Floer-Novikov complexes 371 9 Equivariant transversality 379 10 Group actions on A ∞ -algebra of a Lagrangian submanifold 386 11 G-equivariant A ∞ -bimodule CF * R,l0 (L 0 , L 1 ) 392 12 Equivariant flat vector bundles on G-invariant Lagrangians 396 13 Fukaya-Seidel category of G-Lefschetz fibrations 401 14 Some examples of group actions and mirror symmetry 405 15 CP 2 with a Z/3-action 411 References 416