Irreducible representations of simple Lie algebras by differential operators

Abstract

We describe a systematic method to construct arbitrary highest-weight modules, including arbitrary finite-dimensional representations, for any finite dimensional simple Lie algebra $${\\mathfrak {g}}$$\n g\n . The Lie algebra generators are represented as first order differential operators in $$\\frac{1}{2} \\left( \\dim {\\mathfrak {g}} - \\text {rank} \\, {\\mathfrak {g}}\\right) $$\n \n \n 1\n 2\n \n \n dim\n g\n -\n rank\n \n g\n \n \n variables. All rising generators $$\\mathbf{e}$$\n e\n 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 $$\\mathbf{f}$$\n f\n . We present explicit formulas of this kind for the simple root generators of all classical Lie algebras.