A projective resolution of the symplectic Steinberg module

TL;DR

This paper constructs a projective resolution of the symplectic Steinberg module over number rings and uses it to compute top degree cohomology of certain symplectic groups, extending known resolutions.

Contribution

It provides a new, more involved construction of a projective resolution for the symplectic Steinberg module, similar to Lee--Szczarba's for SL, and applies it to compute cohomology in Euclidean number rings.

Findings

Constructed a projective resolution of the symplectic Steinberg module.

Computed top degree cohomology of principal level-p congruence subgroups.

Extended the understanding of duality and cohomology for symplectic groups over number rings.

Abstract

Borel--Serre proved that for a number ring $R$ with fraction field $K$, the symplectic group $\text{Sp}_{2n}(R)$ is a virtual duality group of degree quadratic in $n$, and that the symplectic Steinberg module $\text{St}^\omega_{2n}(K)$ is its dualizing module. We construct a projective resolution of this symplectic Steinberg module as an $\text{Sp}_{2n}(R)$-representation, that is similar in form to a resolution of Lee--Szczarba for the special linear group, but whose construction is more involved. When $R$ is a Euclidean number ring, we use this resolution to compute the top degree cohomology of principal level-$p$ congruence subgroups of $\text{Sp}_{2n}(R)$, for primes $p \in R$ such that the natural map $R^\times \to (R/(p))^\times$ is surjective.