Conformal higher spin symmetries of 4D massless supermultiplets and osp(L,2M) invariant equations in generalized (super)space

Realization of the conformal higher spin symmetry on the 4D massless field supermultiplets is given. The self-conjugated supermultiplets, including the linearized N=4 super Yang-Mills theory, are considered in some detail. Duality between nonunitary field-theoretical representations and the unitary doubleton-type representations of the 4D conformal algebra su(2,2) is formulated in terms of a Bogolyubov transform. The set of 4D massless fields of all spins is shown to form a representation of sp(8). The results obtained are extended to the generalized superspace invariant under osp(L,2M) supersymmetries. A world line particle interpretation of the free higher spin theories in the osp(2N,2M) invariant (super)space is given. Compatible with unitarity, free equations of motion in the osp(L,2M) invariant (super)space are formulated. A conjecture about the chain of AdS_d+1/CFT_d-->AdS_d/CFT_{d- 1}-->... (where CFT indicates conformal field theory) dualities in the higher spin gauge theories is proposed.