For the hyperelliptic Heisenberg algebra, φ-Verma modules have diagonal Shapovalov form with Legendre norms h_n = 2/(2n+1), are irreducible iff φ is p-admissible, and map explicitly via an intertwiner to polynomials where Sugawara acts as the Legendre operator.
Albino dos Santos, M
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Linear relations among basis elements in the centers of universal central extensions of Der(A) and g⊗A on superelliptic curves match the three-term recurrences of orthogonal polynomials, with explicit proofs for Legendre polynomials when P(x) is quadratic or quartic palindromic.
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A Sugawara-Legendre mechanism for the hyperelliptic Heisenberg algebra
For the hyperelliptic Heisenberg algebra, φ-Verma modules have diagonal Shapovalov form with Legendre norms h_n = 2/(2n+1), are irreducible iff φ is p-admissible, and map explicitly via an intertwiner to polynomials where Sugawara acts as the Legendre operator.
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A partial dictionary between universal central extensions and orthogonal polynomials in the superelliptic Krichever--Novikov setting
Linear relations among basis elements in the centers of universal central extensions of Der(A) and g⊗A on superelliptic curves match the three-term recurrences of orthogonal polynomials, with explicit proofs for Legendre polynomials when P(x) is quadratic or quartic palindromic.