Representations of the $n$ dimensional quantum torus

The $n$-dimensional quantum torus ${\mathcal{O}}_{\mathbf{q}}((F^{\times }{)}^n)$ is defined as the associative $F$-algebra generated by $x_1,\cdots ,x_n$ together with their inverses satisfying the relations $x_i{x}_j=q_{ij}{x}_j{x}_i$, where $\mathbf{q}=(q_{ij})$. We show that the modules that are finitely generated over certain commutative sub-algebras $\mathcal{B}$ are $\mathscr B$-torsion-free and have finite length. We determine the Gelfand-Kirillov dimensions of simple modules in the case when $$\text{Kdim}({\mathcal{O}}_{\mathbf{q}}((F^{\times }{)}^n))=n-1,$$ where $\text{Kdim}$ stands for the Krull dimension. In this case if $M$ is a simple ${\mathcal{O}}_{\mathbf{q}}((F^{\times }{)}^n)$-module then $\text{gk}(M)=1$ or $$\text{gk}(M)\ge \text{gk}({\mathcal{O}}_{\mathbf{q}}((F^{\times }{)}^n))-\text{gk}(\mathcal{Z}({\mathcal{O}}_{\mathbf{q}}((F^{\times }{)}^n)))-1,$$ where $\mathcal Z©$standsforthecenterofanalgebra$C$. We also show that there always exists a simple $F\text{s}A$-module satisfying the above inequality.