McKay conjecture for finite groups in the simplest case G=GL(2,F_p) ( puzzle: Borel knows about cuspidals)

Question

The McKay conjecture and related (Alperin, Issacs-Navarro) are one of the "main problems in the representation theory of finite groups" (G.Navarro pdf). Statement of the McKay conjecture is quite simple:

McKay conjecture: for any finite group $G$ and prime $p$ the following holds:

The number of irreducible representations of G of dimension not divisible by $p$

is equal to

the number of irreducible representations of the normalizer of the Sylow $p$-subgroup $P$ of $G$ of dimension also not divisible by $p$.

i.e.

$$ |\mathrm{Irr}_{p'}(G) | = | \mathrm{Irr}_{p'}(N_G(P)) |. $$

(all representations here are over complex numbers). Notation $\mathrm{Irr}_{p'}(H)$ denotes set of irreducible repsentations of some group $H$ with dimensions not divisible by $p$.

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The conjecture states the equality of the two numbers, but, of course, one may hope that there should be bijection between underlying sets of irreducible representations. So:

Question: What can be a natural bijection in the simplest case $G=\mathrm{GL}(2,\mathbf{F}_p)$ ? (For $p$ same as above).

It is known that there is no choice-free bijection, but, nevertheless there should be some natural family of bijections in order to explain this somehow.

What is puzzling (see more details below) - it is very very straightforward to get HALF of the representations of $G$, from representations of $N_G(P)$ (which is just the Borel subgroup ): it is just the induction from the Borel (=$N_G(P)$) to $G$. But how to get the other half? The other half are the so-called cuspidal representations, which by definition are those which ARE NOT INDUCED from the Borel (=$N_G(P)$) subgroup. Moreover when you do induction you get gluing of irreducible representation of the Borel subgroup by the Weyl group action, so one must make some choice of representatives in the Weyl group orbits to "unglue".

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More details on puzzle:

Let me remind the classification of irreducible representation for $GL(2,F_p)$ (one may look at Garrett page 11, Etingof&K page 69 or refrences in MO271389). And of the Borel (=$N_G(P)$) subgroup.

The puzzling outcome will be that: there should be bijection between (p-1)^2 characters of standard torus (diagonal matrices) and the main part (including cuspidals) of representaions of $GL(2,F_p)$. But that contradicts standard viewpoint - cause cuspidal irreps correspond to characters of the non-split torus, while McKay somehow predicts them from split torus. There seems to be no known way (to me) to get cuspidals from the standard-split torus (=diagonal matrices).

Representations of $\mathrm{GL}(2,\mathbf{F}_p)$ are well-known to come in 4 series:

1) Series 1 - "det" - count = $(p-1)$, dimension = 1

• 1-dimensional representations factoring through the determinant,

2) Series 2 - "regular principal series" - count = $(p-1)(p-2)/2$, dimension = $p+1$

• those induced from the Borel subgroup to $G$ from character of Borel which is "regular" meaning that character has different values on the two generators of the torus,

3) Series 3 - "cuspidal" - count = $(p)(p-1)/2$, dimension = $p-1$

• those that are not induced from Borel and hard to get them

4) Series 4 - "special", count = $(p-1)$, dimension = $p$ ,

• actually those are "irregular principal series".

So the 4th case is not interesting to us in McKay conjecture since, its dimension is $p$ and divisible by $p$.

From cases 2 and 3 we get $(p-1)^2$ irreducible representations and adding those from case 1, we get $(p-1)^2 + (p-1) = p(p-1)$.

Representations of Borel (=$N_G(P)$ - normalizer of Sylow p-subgroup)

Borel = semidirect product of 2-torus and abelian subgroup of unipotent matrices. Representations of semi-direct products easy to describe (see e.g. Etingof&K page 76). So for the particular case of Borel in GL(2):

We have $(p-1)^2$ irreps factoring through the torus (they are one-dimensional), and we have $(p-1)$ non-trivial $(p-1)$-dimensional irreps that are induced from non-trivial characters of the abelian subgroup of unipotent matrices.

So in total we have $(p-1)^2+(p-1) = p(p-1)$.

So we get numerical coincidence $$ p(p-1) = |\mathrm{Irr}_{p'}(GL(2,\mathbf{F}_p) | = | \mathrm{Irr}_{p'}(Borel(2,\mathbf{F}_p)) | = |N_G(P)|. $$ - which is is predicted by the McKay conjecture.

But I do not see how the bijection between the two sets can be made ! (Except it is easy to propose that the $(p-1)$ of type 1 for $\mathrm{GL}(2)$ should correspond to $(p-1)$ non-one-dimensional in irreps of Borel). The problem is that we somehow should get cuspidal from the charactetrs of Borel = characters of the standard torus (diagonal matrices), but there seems to be no known way to do it and moreover cuspidals always correspond to characters of NON-split torus. From high level that is Deligne-Lusztig theory, from down-to-earth considerations that is seen by looking on conjugacy classess as in references above.

Answer 1

Here is an answer of sorts. In this case, by Brauer's First Main Theorem, there is a bijection between $p$-blocks of $G = {\rm GL}(2,p)$ with defect group $P$ and $p$-blocks of $B = N_{G}(P)$ with defect group $P$. Choose a block $\beta$ of $G$ with defect group $P$ and let $\gamma$ be the unique Brauer correspondent of $\beta$ for $B$. The numerical invariants of $\beta$ and $\gamma$ are determined by the theory of blocks of defect one (already developed by Brauer, and later generalized to the cyclic defect group theory by Dade and Green).

In this case, it is (almost) immediate that $B$ has $p-1$ $p$-blocks with defect group $P$ (one for each class of scalar matrices in $B$), and that in each case the all-important invariant, the inertial index $e$ , is $p-1$, being $[N_{G}(P):C_{G}(P)]$ in each case. In fact, one could say that the reason the McKay conjecture holds in this case is because this inertial index is independent of the particular block (of full defect)- possibly together with the fact (also due to Brauer) that in a group whose order is divisible by $p$ but not by $p^{2}$, every irreducible character of degree divisible by $p$ lies in a $p$-block of defect zero.

The general theory of blocks of defect one now gives that the block $\beta$ contains $p$ irreducible characters, all necessarily of degree prime to $p$ as noted above, due to the fact that there are $e + \frac{p-1}{e}$ irreducible characters in a block of defect one, where $e$ is the inertial index of the block.

The same holds within $B = N_{G}(P)$, so that $\gamma$ also contains $p$ irreducible characters, all of degree prime to $p$. Hence in this case, Brauer's First Main Theorem, and the theory of blocks of defect one "explains" why the McKay conjecture holds for $G$.

In some sense it also explains why there is no "canonical" bijection. In a general $p$-block of defect one with inertial index $e$, there are $\frac{p-1}{e}$ "exceptional" irreducible characters, and $e$ "non-exceptional characters". When $e < p-1,$ there is a reasonably natural way to pair the exceptional irreducible characters for the whole group and the exceptional characters for the normalizer of the defect group because of irrationality of character values on non-identity $p$-elements. But when $e = p-1,$ the irrationalities disappear and the distinction between exceptional and non-exceptional characters appears somewhat artificial.