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We develop an elementary method to compute spaces of equivariant maps from ahomogeneous space of a Lie group to a module of this group. The Lie group isnot required to be compact. More generally we study spaces of invariantsections in homogeneous vector bundles, and take a special interest in the casewhere the fibres are algebras. This latter case has a natural global algebrastructure. We classify the resulting automorphic algebras for the case wherethe homogeneous space has compact stabilisers. This work has applications inthe theoretical development of geometric deep learning and also in the theoryof automorphic Lie algebras.

An elementary method to compute equivariant convolutional kernels on homogeneous spaces for geometric deep learning