A Sugawara-Legendre mechanism for the hyperelliptic Heisenberg algebra
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arXiv
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| Natura: | Preprint |
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2026
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| author | Santos, Felipe Albino dos |
| author_facet | Santos, Felipe Albino dos |
| contents | We study the $φ$-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 $M(φ)$ 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, $M(φ)$ is irreducible if and only if $φ$ is $p$-admissible, and this is equivalent to non-degeneracy of the Shapovalov form. Third, there is an explicit intertwiner $Φ\colon M(φ) \to \mathbb{C}[x]$ which sends the free-boson Sugawara zero mode $Ω= -L_0(L_0 + \mathrm{Id}) \in \widetilde{U(\mathcal{H}_m)}$ 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 $\{Ω^r\}_{r \ge 1}$ to $\{L^r\}_{r \ge 1}$. As a companion result, $M(φ)$ is canonically isomorphic to a bosonic Fock space with the Shapovalov form identified with the Fock inner product. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_06090 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Sugawara-Legendre mechanism for the hyperelliptic Heisenberg algebra Santos, Felipe Albino dos Representation Theory 17B67, 17B65, 33C45, 34B24 We study the $φ$-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 $M(φ)$ 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, $M(φ)$ is irreducible if and only if $φ$ is $p$-admissible, and this is equivalent to non-degeneracy of the Shapovalov form. Third, there is an explicit intertwiner $Φ\colon M(φ) \to \mathbb{C}[x]$ which sends the free-boson Sugawara zero mode $Ω= -L_0(L_0 + \mathrm{Id}) \in \widetilde{U(\mathcal{H}_m)}$ 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 $\{Ω^r\}_{r \ge 1}$ to $\{L^r\}_{r \ge 1}$. As a companion result, $M(φ)$ is canonically isomorphic to a bosonic Fock space with the Shapovalov form identified with the Fock inner product. |
| title | A Sugawara-Legendre mechanism for the hyperelliptic Heisenberg algebra |
| topic | Representation Theory 17B67, 17B65, 33C45, 34B24 |
| url | https://arxiv.org/abs/2605.06090 |