Whittaker modules for the Schr\"odinger-Virasoro algebra

Abstract

In this paper, Whittaker modules for the Schr\"odinger-Virasoro algebra $\mathfrak{sv}$ are defined. The Whittaker vectors and the irreducibility of the Whittaker modules are studied. $\mathfrak{sv}$ has a triangular decomposition according to the Cartan algebra $\mathfrak{h}:$ $$\mathfrak{sv}=\mathfrak{sv}^{-}\oplus\mathfrak{h}\oplus\mathfrak{sv}^{+}.$$ For any Lie algebra homomorphism $\psi:\mathfrak{sv}^{+}\to\mathbb{C}$, we can define Whittaker modules of type $\psi.$ When $\psi$ is nonsingular, the Whittaker vectors, the irreducibility and the classification of Whittaker modules are completely determined. When $\psi$ is singular, by constructing some special Whittaker vectors, we find that the Whittaker modules are all reducible. Moreover, we get some more precise results for special $\psi$.