Monstrous Langlands-McKay or what is bijection between conjugacy classes and irreducible representation for sporadic simple groups?

Context: The number of conjugacy classes equals to the number of irreducuble representations (over C) for any finite group. Moreover for the symmetric group and some other groups there is "good bijection" between the two sets (MO153731 ), which in certain sense similar to the Langlands correpondence (MO270916). By the reasons described below it seems (to me) natural to expect such bijection also for sporadic simple groups.

Question 1 Is there some "good" bijection between conjugacy classes and irreducible representation for sporadic simple groups and in particular the Monster ?

There are certain properties of "good" bijection which might be expected for arbitrary finite group (like respect action Our(G), reality/rationality, ... see MO153731).

But for the Monster and sporadic simple groups there are certain specific properties, which might be respected by such "good" bijection:

Question 2 Can the bijection be compatible with "McKay's E8/E6/... observation" ?

McKay observed: The monster has 9 conjugacy classes of elements that can be written as the product of two involutions of type 2A, and their orders are 1, 2, 3, 4, 5, 6, 2, 3, 4. McKay pointed out that these are exactly the numbers appearing on an affine E8 Dynkin diagram giving the linear relation between the simple roots. They are also the degrees of the irreducible representations of the binary icosahedral group (e.g. Borcherds 1998 page 9 bottom ).

Question 2b Are there 9 specific irreducible representations of the Monster which are related to irreps of binary icosahedral group ? They must be bijected to those 9 conjugacy classes ?

There also similar observations for some other sporadic groups. Borcherds writes: "The connection between Dynkin diagrams and 3-dimensional rotation groups is well understood (and is called the McKay correspondence), but there is no known explanation for the connection with the monster. "

The other two properties related to quaternionic irreps and conjugacy classes in the spirit of MO54800 and possible kind of Langlands functoriality for the "happy family".

Motivation for expectation. There are two speculative reasons to expect such bijection one comes from the orbit method, another from the analogy with the Langlands correpondence.

1) For the finite abelian group there is obviosly NO canonical bijecton between irreps and conjugacy classes - they are just dual. On the other pole is symmetric group - there is "good" bijection. It is somewhat similar to the Killing form for the Lie algebras - it is zero for abelian Lie algebras and non-degenerate for simple Lie algebras. Killing form allows to identify $g$ and $g^*$, coadjoint orbits in $g^*$ correspond to irreducible representations, while conjugacy classes by exponential map correspond to adjoint orbits in $g$.

Hence we see that for simple groups we might hope for natural bijection.

2) From the Langlands correspondence point of view, irreducible representations of $G$ correspond to conjugacy classes in Langlands dual group $G^L$ which is sometimes coincide with $G$. Both $G$ and $G^L$ are simple groups, and they are in certain sense of same size up to small differences. However there is no room for sporadic group to have such Langlands-dual partner of same "size": it seems no two sporadic simple groups such that number of conjugacy classes in one equal to number of irreps in the other.

So if we expect that $G->G^L$ preserve simplicity the only choice $G=G^L$ for sporadic groups.

• You write: "Hence we see that for simple groups we might hope for natural bijection." It's not clear what you could expect for simple groups of Lie type, whose character theory has been studied more systematically and uniformly than that of the sporadic simple groups. – Jim Humphreys Jun 5 '17 at 17:51
• @JimHumphreys Is not Lusztig's theory what we might expect ? I have some chain of expectations, and checking some points now... – Alexander Chervov Jun 5 '17 at 19:15
• I'm not sure what I would expect, but I'm also not sure exactly what you mean by "Lustzig's theory" here. He has refined the Deligne-Lusztig construction of generalized characters in many directions, but I don't yet see a natural bijection between classes and irreducible characters. – Jim Humphreys Jun 5 '17 at 21:53
• @JimHumphreys If it is correct: en.wikipedia.org/wiki/… "The representations of G^F are classified using conjugacy classes of the DUAL group of G". I would expect for ANY finite group there exists "Langlands dual" group G^L such that in "FIRST APPROXIMATION" conjugacy classes in G^L biject to irreps of G and certain properties are true. In some cases G=G^L - I hope that for sporadics and some other groups. – Alexander Chervov Jun 6 '17 at 7:07
• @JimHumphreys I would be happy if you shed light on the following question: how far is "true-not-true" in Lusztig's works -- that conjugacy classes in G^L biject to irreps of G ? It seems we must add certain datum to conjugacy classes in G^L like character of normalizer , or it is even more complicated ? – Alexander Chervov Jun 6 '17 at 7:09