On describing invariant subspaces of the space of smooth functions on a homogeneous manifold
Let \(G\) be a Lie group acting on a smooth manifold \(M.\) For every \(g\in G\) and any complex-valued function \(f\) on \(M,\) define \((\pi(g)f)(x)=f(g^{-1}x).\) A locally convex topological vector space \(\mathcal{F}\) of complex-valued functions (or distributions) defined on \(M\) is said to be \(\pi\)-invariant if for any \(f\in\mathcal{F}\) and any \(g\in G,\) \(\pi(g)f\in\mathcal{F}\) and the map \(g\mapsto\pi(g)f\) is continuous from \(G\) to \(\mathcal{F}.\) A vector subspace \(H\subset\mathcal{F}\) is called an invariant subspace if it is closed and \(\pi\)-invariant. The author gives a survey of some results concerning the description of invariant subspaces of function spaces on homogeneous manifolds.
scientific article; zbMATH DE number 2093667