Acta Applicandae Mathematicae 81: 327–338, 2004. © 2004 Kluwer Academic Publishers. Printed in the Netherlands. 327 On Describing Invariant Subspaces of the Space of Smooth Functions on a Homogeneous Manifold S. S. PLATONOV Petrozavodsk State University, 185640 Petrozavodsk, Karelia, Russia. e-mail: platonov@psu.karelia.ru Abstract. Let a Lie group G acts transitively on a manifold M, F a locally convex space consisting of functions on M, and π(g): f (x) → f (g −1 x) the quasi-regular representation of the group G on the topological vector space F . A vector subspace H ⊆ F is called an invariant subspace if it is closed and π -invariant. In the paper we give a survey of some results of the description of invariant subspaces of function spaces on homogeneous manifolods. Mathematics Subject Classifications (2000): Primary: 43A85; secondary: 22E30, 22E46. Key words: invariant subspaces, symmetric spaces, representations of Lie groups, quasi-regular representation. 1. Introduction: The General Problem of Describing Invariant Subspaces Let G be a Lie group that acts transitively on a smooth manifold M. For any g ∈ G and any function f (x) on M, we put (π(g)f )(x) := f (g −1 x). (1.1) A locally convex space F that consists of complex-valued functions (or distribu- tions) on M will be called π -invariant if for any f (x) ∈ F and any g ∈ G we have π(g)f ∈ F and g → π(g)f is a continuous map from G to F . Then the operators π(g)| F (we shall denote them simply by π(g)) define the quasi-regular representation of the group G on the topological vector space F . A vector subspace H ⊆ F is called an invariant subspace if it is closed and π -invariant. One of the main problems of harmonic analysis on Lie groups is to describe all invariant sub- spaces for various Lie groups G, homogeneous manifolds M and various function spaces F distinguished by some restrictions of smoothness and growth. In the present paper we give a survey of some results concerning description of invariant subspaces of function spaces on homogeneous manifolds. In Section 1 we list some results of this kind. Let us note that we do not consider here the most thoroughly studied cases where G is compact or F is a Hilbert space and the quasi-regular representation is unitary. In Sections 2 and 3 we give a description of invariant subspaces in the case where homogeneous manifold is a Riemannian