Irreducible φ-Verma modules for hyperelliptic Heisenberg algebras

We study induced representations of the universal central extension $\widehat{\mathfrak{g}} = (\mathfrak{sl}_2 \otimes R) \oplus \Omega_R^1/dR$, where $R = \mathbb{C}[t^{\pm 1}, u]/(u^2 - p(t))$ is a hyperelliptic coordinate ring and $p(t)$ has degree $r+1$. The center of $\widehat{\mathfrak{g}}$ has dimension $r+1$. Inside $\widehat{\mathfrak{g}}$ sits a hyperelliptic Heisenberg subalgebra $\widehat{\mathfrak{h}}$. A sign function $\varphi \colon \mathbb{Z} \setminus \{0\} \to \{+,-\}$ determines a nonstandard polarization of the imaginary modes, yielding $\varphi$-Verma modules $M_{\widehat{\mathfrak{h}}, \varphi}$ and $\mathcal{M}_\varphi$. Under the specialization $\kappa_1 = \cdots = \kappa_r = 0$ and a $p$-admissibility condition on $\varphi$, we prove: $M_{\widehat{\mathfrak{h}}, \varphi}$ is irreducible if and only if $\kappa_0 \neq 0$, and the same criterion governs $\mathcal{M}_\varphi$ after parabolic induction. For the four-point case $r=1$, we remove the specialization and treat general central characters $(\kappa_0, \kappa_1) \in \mathbb{C}^2$: under $p$-admissibility, $M_{\widehat{\mathfrak{h}}, \varphi}$ is irreducible if and only if $(\kappa_0, \kappa_1) \neq (0,0)$ (Theorem A'). A key ingredient is the closed form $\psi_{mn}(a) = \delta_{m+n, 0}\, \omega_1$ for all $m, n$ when $r=1$, placing the mixed $b^1$-$b$ bracket on the anti-diagonal, independent of the hyperelliptic parameter. We also give a finite checkable criterion for $p$-admissibility via reachable sets in $\mathbb{Z}$. We further describe the weight-space decomposition and formal character of $M_{\widehat{\mathfrak{h}}, \varphi}$, provide a complete structure theorem for the level-zero case, and prove that $p$-admissibility is sharp by constructing explicit reducible modules at nonzero level for non-admissible polarizations.