A Sugawara–Legendre mechanism for the four-point Heisenberg algebra

This paper provides a representation-theoretic explanation for the recent discovery that families of orthogonal polynomials arise in the centres of universal central extensions of superelliptic affine Lie algebras. Working in the Verma modules of the Heisenberg subalgebra, we compute the canonical contravariant form in closed form on a distinguished family of weight vectors and show that this family is the Legendre family, forced by the curve itself. The classical squared norms follow by a rescaling available under a positivity condition on the central charge, and the module is irreducible exactly when that charge is nonzero. The main result identifies a Sugawara element carried, by an explicit intertwiner, to the classical Legendre differential operator. The results are proved for the genus-zero, four-point case.