  The algebra of differential geometry operations on symmetric tensors over
constant curvature manifolds forms a novel deformation of the sl(2,R)
[semidirect product] R^2 Lie algebra. We present a simple calculus for
calculations in its universal enveloping algebra. As an application, we derive
generating functions for the actions and gauge invariances of massive,
partially massless and massless (for both bose and fermi statistics) higher
spins on constant curvature backgrounds. These are formulated in terms of a
minimal set of covariant, unconstrained, fields rather than towers of auxiliary
fields. Partially massless gauge transformations are shown to arise as
degeneracies of the flat, massless gauge transformation in one dimension
higher. Moreover, our results and calculus offer a considerable simplification
over existing techniques for handling higher spins. In particular, we show how
theories of arbitrary spin in dimension d can be rewritten in terms of a single
scalar field in dimension 2d where the d additional dimensions correspond to
coordinate differentials. We also develop an analogous framework for
spinor-tensor fields in terms of the corresponding superalgebra.
