Simple restricted modules over the deformative Schrödinger-Virasoro algebra

This paper investigates simple restricted modules over the deformed Schr\"{o}dinger-Virasoro algebra $\mathcal{G}_{\lambda,\mu}$, which gives a complete classification of them for some $\lambda,\mu\in\mathbb{C}$. More precisely, we provide a systematic construction of these modules, including highest weight modules and Whittaker modules, by inducing simple modules from the positive part's quotient algebras. We prove that any simple restricted $\mathcal{G}_{\lambda,\mu}$-module satisfying certain injective conditions is isomorphic to such an induced module. As an application, we obtain some simple weak $V(c)$-modules over vertex algebras associated to $\mathcal{G}_{\lambda,\mu}$ for some $\lambda,\mu\in\mathbb{C}$. Note that our results include the Schr\"{o}dinger-Virasoro algebra and the deformed $\mathfrak{bms}_3$ algebra as special cases, thereby improving upon some of the previously reported results of [5,Theorem 3.4] and [6,Theorem 2]. This work effectively classifies and generalizes the representation theory of the deformed family.