Finite dimensional representations and subgroup actions on homogeneous spaces

Let H be an ℝ-subgroup of a ℚ-algebraic group G. We study the connection between the dynamics of the subgroup action of H on G/G_ℤ and the representation-theoretic properties of H being observable and epimorphic in G. We show that if H is a ℚ-subgroup then H is observable in G if and only if a certain H orbit is closed in G/G_ℤ; that if H is epimorphic in G then the action of H on G/G_ℤ is minimal, and that the converse holds when H is a ℚ-subgroup of G; and that if H is a ℚ-subgroup of G then the closure of the orbit under H of the identity coset image in G/G_ℤ is the orbit of the same point under the observable envelope of H in G. Thus in subgroup actions on homogeneous spaces, closures of 'rational orbits' (orbits in which everything which can be defined over ℚ, is defined over ℚ) are always submanifolds.