Whittaker vectors in singular Whittaker modules

Abstract

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