Fundamental theorem of K-theory for loop groups over $\mathbb{F}_1$?

Question

As the title says, I would like to know what the fundamental theorem of algebraic K-theory would say over the field with one element. Recall that the fundamental theorem of K-theory provides a decomposition $K_i(R[T,T^{-1}])\cong K_i(R)\oplus K_{i-1}(R)$ for $R$ regular.

Now turn to the philosophical part, the field $\mathbb{F}_1$. For a root system $\Phi$, the natural candidate for the algebraic group of type $\Phi$ over $\mathbb{F}_1$ is the Weyl group $W(\Phi)$ - the formula for the group order is right as noted by Tits, see also several other MO questions on $\mathbb{F}_1$ such as this or this. Now I want to understand the loop group of this group over $\mathbb{F}_1$. The notation would probably be something like $G(\Phi,\mathbb{F}_1[T,T^{-1}])$ and I apologize if anyone is offended by this abuse of notation. I would expect the affine Weyl group for the affine root system $\tilde{\Phi}$ to be a reasonable candidate.

The above philosophical guessing has some consequences, and I would like to know if the following statements have been considered, and if they are true or false. In the following, I specialize to the case of root systems $A_n$ and $\tilde{A}_n$, just for definiteness. In that case $W(A_n)\cong S_{n+1}$ is the symmetric group, and the affine Weyl group $W(\tilde{A}_n)$ is an extension of $S_{n+1}$ by $\mathbb{Z}^{n}$, where $\mathbb{Z}^n$ is identified with the hyperplane $x_1+\cdots+x_{n+1}=0$ in $\mathbb{Z}^{n+1}$ with the permutation action of $S_{n+1}$. Now here are the questions:

1. Are there stabilization theorems for the homology of $W(\tilde{A}_n)$? How do these stabilization theorems compare to those for $S_{n+1}$, in particular is the stable range the same or one less? More generally, has the homology of $W(\tilde{A}_n)$ been computed by any chance (this is loosely related to this MO-question on cohomology of the action of $S_n$ on $T_n$).

2. Provided there are stabilization theorems, have people considered the group completion of $\bigsqcup_n BW(\tilde{A}_n)$ and compared it to the plus-construction space $BW(\tilde{A}_\infty)^+$? A stabilization theorem should imply that the two are weakly equivalent.

3. Does the fundamental theorem of algebraic K-theory hold in this situation? In other words, can we express the homotopy groups of $BW(\tilde{A}_\infty)^+$ in terms of stable homotopy groups of spheres? Maybe $BW(\tilde{A}_\infty)^+$ even splits as two copies of the sphere spectrum (one copy suitably shifted)?

Of course, similar questions can be formulated for the other classical series, and I would be happy to know the answers in these cases as well.

Answer 1

Let $G_n := W(\tilde{A}_{n-1})$. If I understand your description correctly, there is an extension $$1 \to G_n \to S_{n} \wr \mathbb{Z} \overset{sum}\to \mathbb{Z} \to 1$$ and so a $\mathbb{Z}$-Galois cover $BG_n \to B(S_{n} \wr \mathbb{Z})$. There are exterior products $$\mu_{n,m} : G_n \times G_m \to G_{n+m}$$ given by concatenation, and these make $M := \coprod_{n \geq 0} BG_n$ into a homotopy commutative topological monoid. There are similar maps making $M' := \coprod_{n \geq 0} B(S_{n} \wr \mathbb{Z})$ into a homotopy commutative topologial monoid,

By the group-completion theorem of McDuff--Segal, the associated maps $$\mathbb{Z} \times BG_\infty \longrightarrow \Omega B M \quad \quad \mathbb{Z} \times B(S_\infty \wr \mathbb{Z}) \longrightarrow \Omega B M'$$ are both homology equivalences. In fact, by an addendum to the McDuff--Segal work in my paper "Group-Completion", local coefficient systems, and perfection, Quarterly Journal of Mathematics 64 (3) (2013) 795-803, these maps are both in fact acyclic, so yield homotopy equivalences $$\mathbb{Z} \times BG_\infty^+ \simeq\Omega B M \quad \quad \mathbb{Z} \times B(S_\infty \wr \mathbb{Z})^+ \simeq \Omega B M'.$$

The space $B(S_n \wr \mathbb{Z})$ can be modelled as configurations of $n$ points in $\mathbb{R}^\infty$ with labels in $B\mathbb{Z} = S^1$. As such, the group completion $\Omega B M'$ can be approached using Segal's "scanning" technique, and there is a homotopy equivalence $$\Omega BM' \simeq Q(B\mathbb{Z}_+)$$ to the infinite loop space of the spectrum $\mathbf{S}^0 \vee \mathbf{S}^1$.

To get at $\Omega BM$, we can use that the fibration sequence $$BG_\infty \to B(S_\infty \wr \mathbb{Z}) \to B\mathbb{Z}$$ satisfies a condition (e.g. Berrick's) to be plus-constructible, so there is a fibration sequence $$\Omega BM \to Q(B\mathbb{Z}_+) \to B\mathbb{Z}.$$ Thus $\Omega BM$ is the infinite loop space of the spectrum $\mathbf{S}^0 \vee \overline{\mathbf{S}}^1$, where $\overline{\mathbf{S}}^1$ is the 1-connected cover of $\mathbf{S}^1$.

I have carefully avoided discussion of homological stability for the groups $G_n$ (it is not, despite what the question says, necessary (or sufficient) to compare $\mathbb{Z} \times BG_\infty^+$ with $\Omega BM$). I think that homological stability does hold, but it requires a technique not yet in the literature. This technique will hopefully appear in a forthcoming paper.