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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Affine weighted star trees with central parameter k are classified by reducing the positive-semidefinite null-vector condition to the Egyptian-fraction equation sum 1/(r_i+1) = m-k for each fixed (m,k).
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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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Star-Shaped Integral Cartan-Type Matrices and an Egyptian-Fraction Classification of Affine Weighted Trees
Affine weighted star trees with central parameter k are classified by reducing the positive-semidefinite null-vector condition to the Egyptian-fraction equation sum 1/(r_i+1) = m-k for each fixed (m,k).