  Cubic couplings between a complex scalar field and a tower of symmetric
tensor gauge fields of all ranks are investigated on any constant curvature
spacetime of dimension d>2. Following Noether's method, the gauge fields
interact with the scalar field via minimal coupling to the conserved currents.
A symmetric conserved current, bilinear in the scalar field and containing up
to r derivatives, is obtained for any rank r from its flat spacetime
counterpart in dimension d+1, via a radial dimensional reduction valid
precisely for the mass-square domain of unitarity in (anti) de Sitter spacetime
of dimension d. The infinite collection of conserved currents and cubic
vertices are summarized in a compact form by making use of generating functions
and of the Weyl/Wigner quantization on constant curvature spaces.
