  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 $x_ix_j = q_{ij}x_jx_i$, where
$\mathbf q = (q_{ij})$. We show that the modules that are finitely generated
over certain commutative sub-algebras $\mathscr B$ are $\mathscr
B$-torsion-free and have finite length. We determine the Gelfand-Kirillov
dimensions of simple modules in the case when \[ \Kdim(\mathcal O_{\mathbf
q}((F^\times)^n)) = n - 1, \] where $\Kdim$ stands for the Krull dimension. In
this case if $M$ is a simple $\mathcal O_{\mathbf q}((F^\times)^n)$-module then
$ \gk(M) = 1$ or \[ \gk(M) \ge \gk(\mathcal O_{\mathbf q}((F^\times)^n)) -
\gk(\mathcal Z(\mathcal O_{\mathbf q}((F^\times)^n))) - 1,\] where $\mathcal
Z(C)$ stands for the center of an algebra $C$. We also show that there always
exists a simple $F \s A$-module satisfying the above inequality.
