Let *G* be a semisimple algebraic group.

Following work of Matsumoto [1], Brylinski and Deligne [2] constructed a central extension of the functor

*G* : *Rings* → *Groups* by the second algebraic *K*-theory functor.

Plugging in ℂ((t)) into those functors, we get the well known central extension $\widetilde{G\big(\mathbb C((t))}\big)$ of
the loop group *G*(ℂ((t))) by the multiplicative group ℂ*=*K*_{2}(ℂ((t))).

It is interesting to note that the above group comes from an algebraic
group defined over the subfield ℂ of ℂ((t)). Namely, $\widetilde{G\big(\mathbb C((t))}\big)$ = $\widetilde{LG}(\mathbb C)$.

Doing all this with ℚ_{p} instead of ℂ((t)),
we get a central extension $\widetilde{G(\mathbb Q_p)}$ of *G*(ℚ_{p}) by the group *K*_{2}(ℚ_{p}) = **F**_{p}*.
Now, here's an idea: maybe that central extension is defined over... the subfield
**F**_{1} of ℚ_{p}?...

**My questions:**

• Has this been considered before?

• If yes, among all the exitsing notion of "*defined over* **F**_{1}",
which one(s) make this possible?

• If no: is my heuristic argument is convincing?

**References:**

[1] Matsumoto, "Sur les sous-groupes arithmétiques des groupes semi-simples déployés".

[2] Brylinski, Deligne, "Central extensions of reductive groups by $K_2$".

firstquestion should be whether, in some sense (Borger?), $K_2$ is a sheaf on the big Zariski site over the field with one element. Is this known? – Marty May 13 '10 at 18:22