Irreducible $φ$-Verma modules for hyperelliptic Heisenberg algebras

This is the authors' abstract. We don't add key points for this paper.

We study induced representations of the universal central extension g = (sl₂ ⊗ R) ⊕ ΩR¹/dR, where R = C[t± 1, u]/(u² - p(t)) is a hyperelliptic coordinate ring and $p(t)$ has degree $r+1$. The center of g has dimension $r+1$. Inside g sits a hyperelliptic Heisenberg subalgebra h. A sign function φ Z \0\ → \+,-\ determines a nonstandard polarization of the imaginary modes, yielding $φ$-Verma modules M_h, φ and M_φ. Under the specialization κ₁ = ⋯ = κᵣ = 0 and a p-admissibility condition on $φ$, we prove: M_h, φ is irreducible if and only if κ₀ ≠ 0, and the same criterion governs M_φ after parabolic induction. For the four-point case $r=1$, we remove the specialization and treat general central characters (κ₀, κ₁) ∈ C²: under p-admissibility, M_h, φ is irreducible if and only if (κ₀, κ₁) ≠ (0,0) (Theorem A'). A key ingredient is the closed form ψₘₙ(a) = δm+n, 0\, ω₁ for all $m, n$ when $r=1$, placing the mixed b¹-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 Z. We further describe the weight-space decomposition and formal character of M_h, φ, 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.