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Borel = semidirect product of 2-torus and abelian subgroup of unipotent matrices. Its irreps are easy to describe:

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

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):

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.

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

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

So we get numerical coincidence $$ |\mathrm{Irr}_{p'}(GL(2,\mathbf{F}_p) | = | \mathrm{Irr}_{p'}(Borel(2,\mathbf{F}_p)) |. $$ - 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 know 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.

So we get numerical coincidence $$ p(p-1) = |\mathrm{Irr}_{p'}(GL(2,\mathbf{F}_p) | = | \mathrm{Irr}_{p'}(Borel(2,\mathbf{F}_p)) |. $$ - 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.