A Sugawara-Legendre mechanism for the hyperelliptic Heisenberg algebra

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.