The frame-like covariant Lagrangian formulation of bosonic and fermionic mixed-symmetry type higher spin massless fields propagating on the AdS d background is proposed. Higher spin fields are described in terms of gauge p-forms which carry tangent indices representing certain traceless tensor or gamma transversal spinor-tensor representations of the AdS d algebra o(d − 1, 2) (or o(d, 1) for bosonic fields in dS d ). Manifestly gauge invariant Abelian higher spin field strengths are introduced for the general case. We describe the general framework and demonstrate how it works for the mixed-symmetry type fields associated with the three-cell "hook" and arbitrary two-row rectangular tableaux. The manifestly gauge invariant actions for these fields are presented in a simple form. The flat limit is also analyzed.2 Throughout the paper we work within the mostly minus signature and use notations m, n = 0 ÷ d − 1 for world indices, a, b = 0 ÷ d − 1 for tangent Lorentz so(d − 1, 1) vector indices and A, B = 0 ÷ d for tangent (A)dS d (so(d − 1, 2))so(d, 1) vector indices. We also use condensed notations of [19] for a set of symmetric vector indices: a(k) ≡ (a 1 . . . a k ). Upper (lower) indices denoted by the same letter are assumed to be symmetrized as X a Y a ≡ 1 2! (X a1 Y a2 + X a2 Y a1 ) prior contractions.3 Usually, the parameter q in (2.1) satisfies the inequality q ≤ ν, where ν = [ d 2 ] − 1 is the rank of the little group SO(d − 2) for Minkowski space or ν = [ d−1 2 ] is the rank of SO(d − 1) in the case of AdS d , although dual descriptions with larger q are also possible.

Abstract. A constructive procedure is proposed for formulation of linear differential equations invariant under global symmetry transformations forming a semi-simple Lie algebra f. Under certain conditions f-invariant systems of differential equations are shown to be associated with f-modules that are integrable with respect to some parabolic subalgebra of f. The suggested construction is motivated by the unfolded formulation of dynamical equations developed in the higher spin gauge theory and provides a starting point for generalization to the nonlinear case. It is applied to the conformal algebra o(M, 2) to classify all linear conformally invariant differential equations in Minkowski space. Numerous examples of conformal equations are discussed from this perspective. Contents

Covariant lagrangian formulation for free bosonic massless fields of arbitrary mixed-symmetry type in (A)dS d space-time is presented. The analysis is based on the frame-like formulation of higher-spin field dynamics [1] with higher-spin fields described as p-forms taking values in appropriate modules of the (A)dS d algebra. The problem of finding free field action is reduced to the analysis of an appropriate differential complex, with the derivation Q associated with the variation of the action. The constructed action exhibits additional gauge symmetries in the flat limit in agreement with the general structure of gauge symmetries for mixed-symmetry fields in Minkowski and (A)dS d spaces.

We test infinite-dimensional extension of algebra su(k, k) proposed by Fradkin and Linetsky as the candidate for conformal higher spin algebra. Adjoint and twisted-adjoint representations of su(k, k) on the space of this algebra are carefully explored. For k = 2 corresponding unfolded system is analyzed and it is shown to encode Fradkin-Tseytlin equations for the set of all integer spins 1, 2, . . . with infinite multiplicity.

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