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

Given a maximally symmetric $d$-dimensional background with isometry algebra$\mathfrak{g}$, a symmetric and traceless rank-$s$ field $\phi_{a(s)}$satisfying the massive Klein-Gordon equation furnishes a collection of massive$\mathfrak{g}$-representations with spins $j\in \{0,1,\cdots,s\}$. In thispaper we construct the spin-$(s,j)$ projectors, which are operators thatisolate the part of $\phi_{a(s)}$ that furnishes the representation from thiscollection 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 madeearlier in $d=3,4$. We then use these projectors to facilitate a systematicderivation of two-derivative actions with a propagating massless spin-$s$ mode.In addition to reproducing the massless spin-$s$ Fronsdal action, this analysisgenerates new actions possessing higher-depth gauge symmetry. In (A)dS$_d$ wealso derive the action for a partially-massless spin-$s$ depth-$t$ field with$1\leq t \leq s$. The latter utilises the minimum number of auxiliary fields,and corresponds to the action originally proposed by Zinoviev after gaugingaway all St\"{u}ckelberg fields. Some higher-derivative actions are alsopresented, and in $d=3$ are used to construct (i) generalised higher-spinCotton tensors in (A)dS$_3$; and (ii) topologically-massive actions withhigher-depth gauge symmetry. Finally, in four-dimensional $\mathcal{N}=1$Minkowski superspace, we provide closed-form expressions for the analogoussuperprojectors.