Conformal higher spin symmetries of 4D massless supermultiplets andosp(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 $\mathcal{N}=4\mathrm{}$ 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 $\mathrm{su}(2,2)\mathrm{}$ is formulated in terms of a Bogolyubov transform. The set of 4D massless fields of all spins is shown to form a representation of $\mathrm{sp}\left(8\right).$ The results obtained are extended to the generalized superspace invariant under $\mathrm{osp}(L,2M)\mathrm{}$ supersymmetries. A world line particle interpretation of the free higher spin theories in the $osp(2\mathrm{}\mathcal{N},2M)$ invariant (super)space is given. Compatible with unitarity, free equations of motion in the $\mathrm{osp}(L,2M)\mathrm{}$ invariant (super)space are formulated. A conjecture about the chain of ${\mathrm{AdS}}_{d+1\mathrm{}}/{\mathrm{CFT}}_d\to {\mathrm{AdS}}_d/{\mathrm{CFT}}_{d-1}\to \cdot \cdot \cdot$ (where CFT indicates conformal field theory) dualities in the higher spin gauge theories is proposed.