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.

realelements in $G(\mathbb{F}_q)$... That seems a good test case for trying to extend this sort of heuristic. $\endgroup$Conjugacy classes in the Weyl group, Seminar on Algebraic Groups and Related Finite Groups, Lecture Notes in Mathematics 131. $\endgroup$16more comments