Irreducible $\ddot{I}†$-Verma modules for hyperelliptic Heisenberg algebras

Abstract

We study induced representations of the universal central extension $\hat{\mathfrak{g}}=({\mathfrak{sl}}_2\otimes R)\oplus \hat{I}{\text{c◯}}_R^1/dR$, 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 $\hat{\mathfrak{g}}$ has dimension $r+1$. Inside $\hat{\mathfrak{g}}$ sits a hyperelliptic Heisenberg subalgebra $\hat{\mathfrak{h}}$. A sign function $\ddot{I}†:\mathbb{Z}\setminus \{0\}\to \{+,-\}$ determines a nonstandard polarization of the imaginary modes, yielding $\ddot{I}†$-Verma modules $M_{\hat{\mathfrak{h}},\ddot{I}†}$ and ${\mathcal{M}}_{\ddot{I}}†$. Under the specialization $\hat{I}{º}_1=\cdots =\hat{I}{º}_r=0$ and a $p$-admissibility condition on $\ddot{I}†$, we prove: $M_{\hat{\mathfrak{h}},\ddot{I}†}$ is irreducible if and only if $\hat{I}{º}_0\ne 0$, and the same criterion governs ${\mathcal{M}}_{\ddot{I}}†$ after parabolic induction. For the four-point case $r=1$, we remove the specialization and treat general central characters $(\hat{I}{º}_0,\hat{I}{º}_1)\in {\mathbb{C}}^2$: under $p$-admissibility, $M_{\hat{\mathfrak{h}},\ddot{I}†}$ is irreducible if and only if $(\hat{I}{º}_0,\hat{I}{º}_1)\ne (0,0)$ (Theorem A'). A key ingredient is the closed form $\ddot{I}{ˆ}_{mn}(a)=\hat{I}{´}_{m+n,0}\ddot{I}{‰}_1$ 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 $M_{\hat{\mathfrak{h}},\ddot{I}†}$, 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.

Major revision and extension of first version