Irreducible weight modules over the Schr{\"o}dinger Lie algebra in $(n+1)$ dimensional space-time

In this paper, we study weight representations over the Schr{"o}dinger Lie algebra ${\mathfrak{s}}_n$ for any positive integer $n$. It turns out that the algebra ${\mathfrak{s}}_n$ can be realized by polynomial differential operators. Using this realization, we give a complete classification of irreducible weight ${\mathfrak{s}}_n$-modules with finite dimensional weight spaces for any $n$. All such modules can be clearly characterized by the tensor product of ${\mathfrak{s}\mathfrak{o}}_n$-modules, ${\mathfrak{s}\mathfrak{l}}_2$-modules and modules over the Weyl algebra.