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

The dualizing module and top-dimensional cohomology group of $\text{GL}_n(\mathcal{O})$