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We deduce a non-linear commutator higher-spin (HS) symmetry algebra whichencodes unitary irreducible representations of the AdS group -- subject to aYoung tableaux $Y(s_1,\ldots ,s_k)$ with $k\geq 2$ rows -- in a $d$-dimensionalanti-de-Sitter space. Auxiliary representations for a deformed non-linear HSsymmetry algebra in terms of a generalized Verma module, as applied toadditively convert a subsystem of second-class constraints in the HS symmetryalgebra into one with first-class constraints, are found explicitly in the caseof a $k=2$ Young tableaux. An oscillator realization over the Heisenbergalgebra for the Verma module is constructed. The results generalize the methodof constructing auxiliary representations for the symplectic $sp(2k)$ algebraused for mixed-symmetry HS fields in flat spaces \cite{BRbos}. Polynomialdeformations of the $su(1,1)$ algebra related to the Bethe ansatz are studiedas a by-product. A nilpotent BRST operator for a non-linear HS symmetry algebraof the converted constraints for $Y(s_1, s_2)$ is found, with non-vanishingterms (resolving the Jacobi identities) of third order in powers of ghostcoordinates. A gauge-invariant unconstrained reducible Lagrangian formulationfor a free bosonic HS field of generalized spin $(s_1,s_2)$ is deduced.Following the results of \cite{BuchbinderReshetnyak, BRmasscub}, we develop aBRST approach to constructing general off-shell local cubic interactionvertices for irreducible massive higher-spin fields (being candidates formassive particles in the Dark Matter problem). A new reducible gauge-invariantLagrangian formulation for an antisymmetric massive tensor field of spin$(1,1)$ is obtained.

Gauge Invariant Lagrangian Formulations for Mixed Symmetry Higher Spin Bosonic Fields in AdS Spaces