Automorphisms and Verma modules for Generalized Schr\"{o}dinger-Virasoro algebras

Let $\mathbb{F}$ be a field of characteristic 0, $G$ an additive subgroup of $\mathbb{F}$, $\alpha\in \mathbb{F}$ satisfying $\alpha\notin G, 2\alpha\in G$. We define a class of infinite-dimensional Lie algebras which are called generalized Schr\"{o}dinger-Virasoro algebras and use $\mathfrak{gsv}[G,\alpha]$ to denote the one corresponding to $G$ and $\alpha$. In this paper the automorphism group and irreducibility of Verma modules for $\mathfrak{gsv}[G,\alpha]$ are completely determined.