A Sugawara-Legendre mechanism for the hyperelliptic Heisenberg algebra

TL;DR

This paper investigates the structure of $ extphi$-Verma modules over a hyperelliptic Heisenberg algebra, revealing their connection to Legendre polynomials and classical differential operators, with explicit formulas and isomorphisms.

Contribution

It provides explicit descriptions of modules, orthogonal polynomial bases, and intertwiners linking algebraic structures to classical differential operators in the hyperelliptic case.

Findings

Shapovalov form is diagonal in polynomial basis with Legendre norms

Modules are irreducible iff $ extphi$ is $p$-admissible

Explicit intertwiner maps module elements to Legendre polynomials

Abstract

We study the $\varphi$-Verma modules of the Heisenberg subalgebra $\mathcal{H}_m$ of the universal central extension of $\mathfrak{sl}_2 \otimes A_m$, where $A_m$ is the coordinate ring of the superelliptic curve $u^m = P(t)$, and ask how the orthogonal polynomial families that arise in the centre relations are controlled by the module theory of $\mathcal{H}_m$. Our main results are proved unconditionally for the hyperelliptic case $m=2$, $r=1$; corresponding statements for $m \ge 3$ are recorded as conjectures. In the hyperelliptic case we prove three theorems. First, the canonical contravariant (Shapovalov) form on $M(\varphi)$ is diagonal in the polynomial basis $\{\tilde{P}_n\}_{n \ge 0}$ determined by the cocycle, with Legendre squared norms $h_n = 2/(2n+1)$. Second, $M(\varphi)$ is irreducible if and only if $\varphi$ is $p$-admissible, and this is equivalent to non-degeneracy of the Shapovalov form. Third, there is an explicit intertwiner $\Phi \colon M(\varphi) \to \mathbb{C}[x]$ which sends the free-boson Sugawara zero mode $\Omega = -L_0(L_0 + \mathrm{Id}) \in \widetilde{U(\mathcal{H}_m)}$ to the classical Legendre differential operator $L = (1-x^2)\partial_x^2 - 2x\partial_x$, the level-$n$ image of the highest-weight vector to the Legendre polynomial $P_n(x)$, and the Casimir tower $\{\Omega^r\}_{r \ge 1}$ to $\{L^r\}_{r \ge 1}$. As a companion result, $M(\varphi)$ is canonically isomorphic to a bosonic Fock space with the Shapovalov form identified with the Fock inner product.