Irreducible $φ$-Verma modules for hyperelliptic Heisenberg algebras
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2017
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| author | Santos, Felipe Albino dos |
| author_facet | Santos, Felipe Albino dos |
| contents | We study induced representations of the universal central extension $\widehat{\mathfrak{g}} = (\mathfrak{sl}_2 \otimes R) \oplus Ω_R^1/dR$, 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 $φ\colon \mathbb{Z} \setminus \{0\} \to \{+,-\}$ determines a nonstandard polarization of the imaginary modes, yielding $φ$-Verma modules $M_{\widehat{\mathfrak{h}}, φ}$ and $\mathcal{M}_φ$. Under the specialization $κ_1 = \cdots = κ_r = 0$ and a $p$-admissibility condition on $φ$, we prove: $M_{\widehat{\mathfrak{h}}, φ}$ is irreducible if and only if $κ_0 \neq 0$, and the same criterion governs $\mathcal{M}_φ$ after parabolic induction. For the four-point case $r=1$, we remove the specialization and treat general central characters $(κ_0, κ_1) \in \mathbb{C}^2$: under $p$-admissibility, $M_{\widehat{\mathfrak{h}}, φ}$ is irreducible if and only if $(κ_0, κ_1) \neq (0,0)$ (Theorem A'). A key ingredient is the closed form $ψ_{mn}(a) = δ_{m+n, 0}\, ω_1$ 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 $M_{\widehat{\mathfrak{h}}, φ}$, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1709_05663 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Irreducible $φ$-Verma modules for hyperelliptic Heisenberg algebras Santos, Felipe Albino dos Representation Theory 17B10, 17B65, 17B68, 17B22 We study induced representations of the universal central extension $\widehat{\mathfrak{g}} = (\mathfrak{sl}_2 \otimes R) \oplus Ω_R^1/dR$, 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 $φ\colon \mathbb{Z} \setminus \{0\} \to \{+,-\}$ determines a nonstandard polarization of the imaginary modes, yielding $φ$-Verma modules $M_{\widehat{\mathfrak{h}}, φ}$ and $\mathcal{M}_φ$. Under the specialization $κ_1 = \cdots = κ_r = 0$ and a $p$-admissibility condition on $φ$, we prove: $M_{\widehat{\mathfrak{h}}, φ}$ is irreducible if and only if $κ_0 \neq 0$, and the same criterion governs $\mathcal{M}_φ$ after parabolic induction. For the four-point case $r=1$, we remove the specialization and treat general central characters $(κ_0, κ_1) \in \mathbb{C}^2$: under $p$-admissibility, $M_{\widehat{\mathfrak{h}}, φ}$ is irreducible if and only if $(κ_0, κ_1) \neq (0,0)$ (Theorem A'). A key ingredient is the closed form $ψ_{mn}(a) = δ_{m+n, 0}\, ω_1$ 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 $M_{\widehat{\mathfrak{h}}, φ}$, 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. |
| title | Irreducible $φ$-Verma modules for hyperelliptic Heisenberg algebras |
| topic | Representation Theory 17B10, 17B65, 17B68, 17B22 |
| url | https://arxiv.org/abs/1709.05663 |