A partial dictionary between universal central extensions and orthogonal polynomials in the superelliptic Krichever--Novikov setting

Abstract

Let $m \geq 2$, let $P(x) \in \mathbb{C}[x]$ have simple roots, and let $A = \mathbb{C}[x^{\pm 1},\,u \mid u^m = P(x)]$ be the coordinate ring of the associated superelliptic curve. The derivation algebra $\mathrm{Der}(A)$ and the current algebra $\mathfrak{g}\otimes A$ (for $\mathfrak{g}$ a simple Lie algebra) each admit a universal central extension whose center is multi-dimensional and carries linear algebraic relations among its basis elements. We establish a systematic dictionary between these relations and families of orthogonal polynomials in the parameter $a$ encoding the branch locus of $P$. The dictionary has three canonical entries: (1)~the basis reduction relations in the center of $\widehat{\mathrm{Der}(A)}$ are exactly the three-term recurrence of an orthogonal polynomial family; (2)~the generating function of the center satisfies the Sturm--Liouville ODE of that family; (3)~the mixed-sector $2$-cocycle equals the Legendre antiderivative $(P_{n-1}(a)-P_{n+1}(a))/(2n+1)$ in the quadratic case. We prove the dictionary completely for $P(x)=x^2-2ax+1$ (Legendre polynomials) and for the quartic palindromic case $P(x)=x^4-2ax^2+1$. In the quadratic case, palindromic symmetry forces the recurrence to be the Legendre three-term recurrence; in the quartic case, the odd sector is Legendre and the even sector satisfies a two-component recurrence with palindromic coefficients. We conjecture this pattern -- palindromic $P$ forcing symmetric recurrence coefficients -- holds in all even degrees. The same dictionary governs the K\"{a}hler side $\Omega^1_A/dA$: all sectors reduce to the sector-$1$ family at a rescaled parameter, and the recurrence and ODE entries are canonical while the mixed-cocycle entry is partially choice-dependent.