Spin-(s, j) projectors and gauge-invariant spin-s actions in maximally symmetric backgrounds

Given a maximally symmetric d-dimensional background with isometry algebra g, a symmetric and traceless rank-s field ϕ_a(s) satisfying the massive Klein-Gordon equation furnishes a collection of massive g-representations with spins j ∈ {0, 1, · · ·, s}. In this paper we construct the spin-(s, j) projectors, which are operators that isolate the part of ϕ_a(s) that furnishes the representation from this collection carrying spin j. In the case of an (anti-)de Sitter ((A)dS_d) background, we find that the poles of the projectors encode information about (partially-)massless representations, in agreement with observations made earlier in d = 3, 4. We then use these projectors to facilitate a systematic derivation of two-derivative actions with a propagating massless spin-s mode. In addition to reproducing the massless spin-s Fronsdal action, this analysis generates new actions possessing higher-depth gauge symmetry. In (A)dS_d we also derive the action for a partially-massless spin-s depth-t field with 1 ≤ t ≤ s. The latter utilises the minimum number of auxiliary fields, and corresponds to the action originally proposed by Zinoviev after gauging away all Stückelberg fields. Some higher-derivative actions are also presented, and in d = 3 are used to construct (i) generalised higher-spin Cotton tensors in (A)dS₃; and (ii) topologically-massive actions with higher-depth gauge symmetry. Finally, in four-dimensional N = 1 Minkowski superspace, we provide closed-form expressions for the analogous superprojectors.