Totally real representations and real function spaces

Let G be a locally compact group.The notion of "totally real" unitary representation of G is defined and investigated in §1.In particular, if K is a compact subgroup of G, it is shown that every closed G-invariant subspace of L 2 (G/K) is spanned by real-valued functions if, and only if, KgK=Kg~xK for every geG.In §2 the coset space X=G/K is specialized to a Riemannian symmetric space, where the double coset condition is replaced by a simple Weyl group condition.