Weight modules with a finite dimensional weight space over truncated Virasoro algebras

Abstract

Truncated Virasoro algebras can be viewed as quotient algebras of the loop–Virasoro algebra. In this paper, we prove that any weight module M with a finite dimensional weight space over any truncated Virasoro algebra \documentclass[12pt]{minimal}\begin{document}${\cal L}(f)={\rm {Vir}}\otimes (\mathbb {C}[t^{\pm 1}]/f\mathbb {C}[t^{\pm 1}])$\end{document}L(f)= Vir ⊗(C[t±1]/fC[t±1]) has a Harish-Chandra submodule, where Vir is the Virasoro algebra and f is a nonzero and not invertible Laurent polynomial in \documentclass[12pt]{minimal}\begin{document}$\mathbb {C}[t^{\pm 1}]$\end{document}C[t±1]. More precisely, if M has a nontrivial upper bounded Vir-submodule, then it has a highest weight \documentclass[12pt]{minimal}\begin{document}${\cal L}(f)$\end{document}L(f)-submodule; if M has a nontrivial lower bounded Vir-submodule, then it has a lowest weight \documentclass[12pt]{minimal}\begin{document}${\cal L}(f)$\end{document}L(f)-submodule; if M is nontrivial over \documentclass[12pt]{minimal}\begin{document}${\cal L}(f)$\end{document}L(f) and does not contain any nontrivial upper bounded nor any nontrivial lower bounded Vir-submodules, then it has a nontrivial Harish-Chandra \documentclass[12pt]{minimal}\begin{document}${\cal L}(f)$\end{document}L(f)-submodule S such that \documentclass[12pt]{minimal}\begin{document}$\dim S_{\lambda }=1$\end{document}dimSλ=1 for all λ ∈ supp(S)∖{0}. As a corollary, any irreducible weight module with a finite dimensional weight space over \documentclass[12pt]{minimal}\begin{document}${\cal L}(f)$\end{document}L(f) is a Harish-Chandra module.