Representations and the Jacobian Conjecture

In this paper we apply the representation theory of the Lie algebra image to the problem of describing Hessian nilpotent polynomials, which are important in the theory of the Jacobian Conjecture. In the two variable case we describe them as the maximal and minimal weight vectors of the irreducible representations of image. For the first time this gives a characterization of the Hessian nilpotent polynomials in terms of linear differential operators.