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This work studies equivariant linear second order elliptic operators P on a connected noncompact manifold X with a given action of a group G. The action is assumed to be cocompact, meaning that GV=X for some compact subset V of X. The aim is to study the structure of the convex cone of all positive solutions of Pu=0. It turns out that the set of all normalized positive solutions which are also eigenfunctions of the given G -action can be realized as a real analytic submanifold *G[0 of an appropriate topological vector space *H. When G is finitely generated, *H has finite dimension, and in nontrivial cases *G[0 is the boundary of a strictly convex body in *H. When G is nilpotent, any positive solution u can be represented as an integral with respect to some uniquely defined positive Borel measure over *G[0. Lin and Pinchover also discuss related results for parabolic equations on X and for elliptic operators on noncompact manifolds with boundary.