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

Let G be the planar Galilean conformal algebra and G˜ its universal central extension. Then (resp. G˜) admits a triangular decomposition: G=G+?G0?G? G˜=G˜+?G˜0?G˜?). In this paper, we study generic Whittaker G-modules G˜-modules) of type ?, where ?:G+?G˜+?C is Lie homomorphism. We classify isomorphism classes modules. Moreover, show that module ? simple if only nonsingular. For nonsingular case, completely determine vectors in singular concretely construct some proper submodules