Subalgebras of ₙ and Jacobi polynomials

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We classify the subalgebras of the general Lie conformal algebra ₙ that act irreducibly on [∂]ⁿ and that are normalized by the sl₂--part of a Virasoro element. The problem turns out to be closely related to classical Jacobi polynomials Pₙ⁽⁻σ,σ⁾, σ∈. 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.