$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Sp{Sp}$Suppose that $F/Q$ is a number field.


Using automorphic forms, Borel computed the ($R$-) stable cohomology of $\SL\_n(O\_F)$, and as a result, computed $K\_i(O\_F)\otimes Q$. Nowadays, using work of Suslin, Voevodksy, Rost, etc, one has an almost "complete" understanding of the actual integral groups $K\_i(Z)$, say, modulo Vandiver's conjecture. This does not *directly* give the stable cohomology of $\SL\_n$, because one set of groups is computing homotopy and the other is computing cohomology, but let us not worry about that distinction for the moment.


Borel also computed the ($R$-) stable cohomology of $\Sp\_{2n}(O\_F)$.
My question, loosely speaking, is whether one can describe the stable integral cohomology of $\Sp\_{2n}(Z)$ in as detailed away as the algebraic K-groups of $Z$ "describe" the integral cohomology of $\SL\_n(Z)$.




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Let me summarize some of what I have found out (following up some of the answers below), mostly though emails from experts.


For affine objects, which certainly includes $Z$, K-theory is about the monoidal category
$P(A)$ of projective finitely generated A-modules, and
Hermitian K-theory is about the monoidal category
$P(A)\_h$ of objects in $P(A)$ equipped with a non-degenerate
symmetric (or skew-symmetric) form. One of the issues with computing or working with such a theory over $Z$ is that irritating issues arise in characteristic $2$, as one might expect with quadratic forms present. It seems that one might be in good shape to understand the groups $K^h\_i(Z[1/2])$. For usual K-theory, there is an excision formula relating $K\_i(Z)$ to $K\_i(Z[1/2])$ and $K\_i(F\_2)$. The latter group is "easy" (or at least was computed by Quillen).


Of interest to me in $K\_i(Z)$ are the Soulé classes. The next thing I will be thinking about is whether Soule classes can give rise to elements in the stable cohomology of $\Sp\_2n(Z)$.


Andy said some very interesting things, but I will probably be awarding the bounty to Oscar, since the paper he linked to was more directly relevant to what I was trying to find out. (Sorry Andy...but it looks like you have lots of reputation anyway!)



Stable homology of arithmetic groups
The algebraic K-groups of Z are to the homology of SLn(Z) as the **Hermitian K-groups** of Z are to the homology of Sp2g(Z). There is a paper by Berrick and Karoubi [here](https://faculty.math.illinois.edu/K-theory/0649/BerrickK.pdf "Hermitian K-theory of the integers") in which they discuss, and make some calculations of, the Hermitian K-theory of Z and Z[1/2].


