The dualizing module and top-dimensional cohomology group of GL _n(O)

For a number ring O, Borel and Serre proved that SL _n(O) is a virtual duality group whose dualizing module is the Steinberg module. They also proved that GL _n(O) is a virtual duality group. In contrast to SL _n(O) , we prove that the dualizing module of GL _n(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 GL _n(O) in its virtual cohomological dimension.