Whittaker vectors in singular Whittaker modules

Let g be a complex semisimple Lie algebra with Borel subalgebra b and corresponding nilradical n. We show that singular Whittaker modules M are simple if and only if the space Wh M of Whittaker vectors is 1-dimensional. For arbitrary locally n-finite g-modules V, an immediate corollary is that the dimension of Wh V is bounded by the composition length of V.