A projective resolution of the symplectic Steinberg module

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 ${\text{St}}_{2n}^{\omega }(K)$ 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 $R^{\times }\to (R\mathrm{/}(p){)}^{\times }$ is surjective.