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We use Arkhipov's twisting functors to show that the universal enveloping algebra of a semi-simple complex finite-dimensional Lie algebra surjects onto the space of ad-finite endomorphisms of the simple highest weight module L(λ), whose highest weight is associated (in the natural way) with a subset of simple roots and a simple root in this subset. This is a new step towards a complete answer to a classical question of Kostant. We also show how one can use the twisting functors to reprove the classical results related to this question.doi:10.1017/s0017089505002776 fatcat:gylaseb2bjeovc3nu5ojjrsany