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 L2(G/K) is spanned by real-valued functions if, and only if, KgK=Kg~K for every geG. In §2 the coset space X=G/K is specialized to a Riemannian symmetric space, where th...
