  For a number ring $\mathcal{O}$, Borel and Serre proved that
$\text{SL}_n(\mathcal{O})$ is a virtual duality group whose dualizing module is
the Steinberg module. They also proved that $\text{GL}_n(\mathcal{O})$ is a
virtual duality group. In contrast to $\text{SL}_n(\mathcal{O})$, we prove that
the dualizing module of $\text{GL}_n(\mathcal{O})$ is sometimes the Steinberg
module, but sometimes instead is a variant that takes into account a sort of
orientation. Using this, we obtain vanishing and nonvanishing theorems for the
cohomology of $\text{GL}_n(\mathcal{O})$ in its virtual cohomological
dimension.
