On representations of Higher Spin symmetry algebras for mixed-symmetry HS fields on AdS-spaces. Lagrangian formulation

We derive non-linear commutator HS symmetry algebra, which encode unitary irreducible representations of AdS group subject to Young tableaux $Y(s_1,\mathrm{.}\mathrm{.}\mathrm{.},s_k)$ with $k\ge 2$ rows on $d$-dimensional anti-de-Sitter space. Auxiliary representations for specially deformed non-linear HS symmetry algebra in terms of generalized Verma module in order to additively convert a subsystem of second-class constraints in the HS symmetry algebra into one with first-class constraints are found explicitly for the case of HS fields for $k=2$ Young tableaux. The oscillator realization over Heisenberg algebra for obtained Verma module is constructed. The results generalize the method of auxiliary representations construction for symplectic $sp(2k)$ algebra used for mixed-symmetry HS fields on a flat spaces and can be extended on a case of arbitrary HS fields in AdS-space. Gauge-invariant unconstrained reducible Lagrangian formulation for free bosonic HS fields with generalized spin $(s_1,s_2)$ is derived.