  We develop an elementary method to compute spaces of equivariant maps from a
homogeneous space $G/H$ of a Lie group $G$ to a module of this group. The Lie
group is not required to be compact. More generally, we study spaces of
invariant sections in homogeneous vector bundles, and take a special interest
in the case where the fibres are algebras. These latter cases have a natural
global algebra structure. We classify these automorphic algebras for the case
where the homogeneous space has compact stabilisers. This work has applications
in the theoretical development of geometric deep learning and also in the
theory of automorphic Lie algebras.
