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 $\varphi :{\mathcal{G}}^{+}={\tilde{\mathcal{G}}}^{+}⟶\mathbb{C}$ is a Lie algebra homomorphism. We classify the isomorphism classes of universal and generic Whittaker modules. Moreover, we show that a generic Whittaker modules of type $\varphi$ is irreducible 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.

26 pages.Some proofs are simplified