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Let $K$ be a number field with ring of integers $R = \mathcal{O}_K$. We showthat if $R$ is not a principal ideal domain, then the symplectic group$\operatorname{Sp}_{2n}(R)$ has non-trivial rational cohomology in its virtualcohomological dimension. This demonstrates a sharp contrast to the situationwhere $R$ is Euclidean. To prove our result, we study the symplectic Steinbergmodule, i.e. the top-dimensional homology group of the spherical buildingassociated to $\operatorname{Sp}_{2n}(K)$. We show that this module is notgenerated by integral apartment classes.

Top-degree rational cohomology in the symplectic group of a number ring