Irreducible representations of simple Lie algebras by differential operators

We describe a systematic method to construct arbitrary highest-weight modules, including arbitrary finite-dimensional representations, for any finite dimensional simple Lie algebra $${\mathfrak {g}}$$ . The Lie algebra generators are represented as first order differential operators in $$\frac{1}{2} \left( \dim {\mathfrak {g}} - \text {rank} \, {\mathfrak {g}}\right) $$ variables. All rising generators $$\mathbf{e}$$ are universal in the sense that they do not depend on representation, the weights enter (in a very simple way) only in the expressions for the lowering operators $$\mathbf{f}$$ . We present explicit formulas of this kind for the simple root generators of all classical Lie algebras.