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

Abstract

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

18 pages, 1 figure; v2: corrected typos and improved the exposition; v3: final version, to appear in Selecta Math. (N.S.)

References in corpus (1)

• On the top-dimensional cohomology of arithmetic Chevalley groups