Whittaker modules for the planar Galilean conformal algebra and its central extension

Abstract

Let $\mathcal{G}$ be the planar Galilean conformal algebra and $\tilde{\mathcal{G}}$ be its universal central extension. Then $\mathcal{G}$ (resp. $\tilde{\mathcal{G}}$) admits a triangular decomposition: $\mathcal{G}={\mathcal{G}}^{+}\oplus {\mathcal{G}}^0\oplus {\mathcal{G}}^{-}$ (resp. $\tilde{\mathcal{G}}={\tilde{\mathcal{G}}}^{+}\oplus {\tilde{\mathcal{G}}}^0\oplus {\tilde{\mathcal{G}}}^{-}$). In this paper, we study universal and generic Whittaker $\mathcal{G}$-modules (resp. $\tilde{\mathcal{G}}$-modules) of type $\varphi ,$ where $\varphi :{\mathcal{G}}^{+}\cong {\tilde{\mathcal{G}}}^{+}\to \mathbb{C}$ is a Lie algebra homomorphism. We classify the isomorphism classes of universal and generic Whittaker modules. Moreover, we show that a generic Whittaker module of type $\varphi$ is simple if and only if $\varphi$ is nonsingular. For the nonsingular case, we completely determine the Whittaker vectors in universal and generic Whittaker modules. For the singular case, we concretely construct some proper submodules of generic Whittaker modules.

Keywords:

• (Non-)singular type
• planar Galilean conformal algebra and its central extension
• simple module
• Whittaker module
• Whittaker vector

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 17B10
• 17B35
• 17B65
• 17B68

Acknowledgments

Y.F. Yao is grateful to Professor Kaiming Zhao for stimulating discussion and helpful suggestion which improves the manuscript.

Additional information

Funding

This work is supported by National Natural Science Foundation of China (Grant Nos. 12271345 and 12071136).