@FedericoPoloni At least today the participants and speakers of the meeting seem to be from all over the world. Doesn't that make it an international conference?

@DaveBenson: Indeed. A nice book that discusses this part of the history in more detail: T. Hawkins, Emergence of the theory of Lie groups. An essay in the history of mathematics 1869-1926. Sources and Studies in the History of Mathematics and Physical Sciences. Springer-Verlag, New York, 2000. xiv+564 pp doi.org/10.1007/978-1-4612-1202-7

I suppose 51840 comes from the order of $O_6^{-}(2)$, which is a group that Jordan studied for other reasons. And $W(E_6) \cong O_6^{-}(2)$.

@kindasorta: Sorry, that was a mistake - it holds for all $k \geq 1$.

I have no interest in the Langlands program, but the dual root datum is relevant in the representation theory of finite groups of Lie type (Deligne-Lusztig theory), where you can define the analogue of ${}^L G$ over a finite field. I'm not sure if this is any more elementary, but because of my own interests it is more approachable to me..

With what I wrote $vv' = f(v,v') \prod_k v_k^{i_k+i_k'} \prod_k w_k^{j_k+j_k'}$ in $P$, isn't that correct?

Thanks, I see - then just use $gnu(4n) = \sum_{T} gnu_T(4n)$.

Can you clarify how the result follows from result for $\mathcal{S}$? Namely I wonder if there is an issue with groups $T \in \mathcal{S}'$ such that $|T| \mid 4n$, but $|T| \nmid n$.