Year: 2025
Type: article
Abstract: Let K be a number field with ring of integers $$R = \mathcal {O}_K$$ R = O K . We show that if R is not a principal ideal domain, then the symplectic group $$\operatorname {Sp}_{2n}(R)$$ Sp 2 n ( 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 $$\operatorname {Sp}_{2n}(K)$$ Sp 2 n ( K ) . We show that this module is not generated by integral apartment classes. Both of these results follow from a vanishing theorem for homology with Steinberg coefficients.
Source: Selecta Mathematica
Authors: Benjamin Brück, Zachary Himes
Institutions: University of Münster, University of Michigan
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
OpenAlex ID: W4411080894