Subalgebras of gc N and Jacobi Polynomials

Abstract We classify the subalgebras of the general Lie conformal algebra $\text{g}{{\text{c}}_{N}}$ that act irreducibly on $\mathbb{C}\,{{\left[ \partial \right]}^{N}}$ and that are normalized by the $\text{s}{{\text{l}}_{2}}$ -part of a Virasoro element. The problem turns out to be closely related to classical Jacobi polynomials $P_{n}^{\left( -\sigma ,\sigma \right)}$ , $\sigma \,\in \,C$ . The connection goes both ways—we use in our classification some classical properties of Jacobi polynomials, and we derive from the theory of conformal algebras some apparently new properties of Jacobi polynomials.