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We study three related topics in representation theory of classical Liesuperalgebras. The first one is classification of primitive ideals, i.e.annihilator ideals of simple modules, and inclusions between them. The secondtopic concerns Arkhipov's twisting functors on the BGG category O. The thirdtopic addresses deformed orbits of the Weyl group. These take over the role ofthe usual Weyl group orbits for Lie algebras, in the study of primitive idealsand twisting functors for Lie superalgebras.

Primitive ideals, twisting functors and star actions for classical Lie superalgebras