We develop an elementary method to compute spaces of equivariant maps from a homogeneous space of a Lie group 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. This latter case has a natural global algebra structure. We classify the resulting 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.

翻译：我们提出一种初等方法，用于计算从李群的齐性空间到该群模空间的等变映射空间。该方法不要求李群是紧致的。更一般地，我们研究齐性向量丛中的不变截面空间，并特别关注纤维为代数的情形。后者具有自然的全局代数结构。我们分类了齐性空间具有紧致稳定子群时的自守代数。本工作对几何深度学习的理论发展以及自守李代数理论均具有应用价值。