TITLE: Analogies supporting heuristic: Weyl groups = algebraic groups over field with one element?
QUESTION [32 upvotes]: There is well-known heuristic that Weyl groups are  reductive algebraic groups over "field with one element".
Probably the best known analogy supporting that heuristic is the limit $q\to1$ for number of elements in $G(F_q)$ - for appropriate "m" it holds:
$$ \lim_{q\to1} \frac { |G(F_q) | } { (q-1)^m}  = |Weyl~Group~of~G| $$
For example: $|GL(n,F_q)|= [n]_q! (q-1)^{n}q^{n(n-1)/2} $ so divided by $(q-1)^n$ one gets $[n]_q! q^{n(n-1)/2} $ and at the limit $q\to1$, one gets $n!$  which is the size of $S_n$ (Weyl group for GL(n)).
(For other groups see Lorscheid 2009 page 2 formula 1).
Question What are the other analogies supporting heuristics: Weyl groups = algebraic groups over field with one element ?
Subquestion once googling papers on F_1, I have seen quite an interesting  analogy from representation theory point of view - it was some fact about induction from diagonal subgroups of symmetric groups $S_{d_1}\times ... \times S_{d_k} \subset S_n$ where $\sum d_i = n$ and similar fact for $GL(n,F_q)$ which was due to  Steinberg or Springer or Carter (cannot remember).
But I cannot google it again and cannot remember the details :(  (Tried quite a lot - I was sure it was on the first or second page of Soule's paper on F_1 - but it is not there, neither many other papers).

Knowing that total element count is okay, we may ask  about counting
elements with certain properties - like: m-tuples of commuting elements (MO271752),
involutions, elements of order $m$, whatever  ...
From answer MO272059 one knows that there are certain analogies for such counting, however it seems the limits $q\to1$ are not quite clear.
Question 2 Is there any analogy for counting elements with some reasonable conditions ? Hope to see that count for $G(F_q)$ (properly normalized) in the limit $q\to1$    gives answer for Weyl group.

REPLY [5 votes]: [The following comment is too long for the comment box.]
On the subquestion: Zelevinsky's Representations of finite classical groups - a Hopf algebra approach (LNM) may be relevant to what you're thinking of. Zelevinsky builds two Hopf algebras: the first coming from induction and restriction of (complex) representations of the symmetric groups along the inclusion $S_n\times S_m\to S_{n+m}$, the second using parabolic induction and restriction (again, of complex representations) for finite general linear groups along the inclusion $GL_n(\mathbb F_q)\times GL_m(\mathbb F_q)\to GL_{n+m}(\mathbb F_q)$. Zelevinsky shows that the second algebra is a tensor product of copies of the first, with one copy for each pair $(n,\pi)$ where $\pi$ is a cuspidal representation of $GL_n(\mathbb F_q)$. 
Here's an attempt, possibly completely misguided, to extract from this an analogy that would be relevant to your question. (I don't know how this relates to the existing literature on $\mathbb{F}_1$. Sorry if I am repeating something that is well known.) If we identify $S_n=GL_n(\mathbb F_1)$, Zelevinsky's result might be interpreted metaphorically as saying that ''the only cuspidal representation of $GL_n(\mathbb F_1)$ is the trivial representation of the trivial group''. Since cuspidal representations of $GL_n(\mathbb F_q)$ are associated to characters of anisotropic tori, and since (I suppose?) the ''group of $\mathbb F_1$-points of a torus over $\mathbb F_1$'' is always the trivial group, this makes some kind of sense.