Temel Kavramlar
The paper develops an elementary method to compute the space of G-equivariant maps from a homogeneous space G/H to a module V of the group G, without requiring G to be compact. This has applications in the theoretical development of geometric deep learning and the theory of automorphic Lie algebras.
Özet
The paper focuses on computing the space of G-equivariant maps from a homogeneous space G/H to a module V of the group G. This is motivated by recent developments in geometric deep learning, where equivariant convolutional kernels are important, as well as the theory of automorphic Lie algebras.
The key steps of the method are:
Choose a base point x0 in the homogeneous space G/H and determine the stabilizer subgroup H = Gx0.
Find a map f: G/H → G such that f(x)x0 = x.
Compute the space of H-invariants V^H.
The G-equivariant maps G/H → V are then given by the maps x → f(x)v, where v ranges over a basis of V^H.
The paper also generalizes this to the case where the module V has the structure of a G-algebra, leading to the concept of automorphic algebras. Several examples are provided, including the hyperbolic plane, the sphere, and hyperbolic 3-space.