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Due to security issues and the fact that I want to use only secure connections (also for my own MathJax-Server-Connections) -- new versions of Firefox for example do not automatically allow mixed connection-websites with https and http protocols -- you may experience some problems with \(La\TeX\), until I have time to implement "the" elegant solution. The non-elegant one for you now may simply be: go to math \dot jmcrv \dot org and accept the self-signed certificate there. Then you will see always nice \(La\TeX\)!. My apologies for the inconveniences. The problem will be solved soon! Thank you.

In the lecture, we constructed irreducible representations of \(GL_2(\mathbb{F}_q)\) corresponding to characters \(\chi=(\chi_1,\chi_2)\) of \(T\):

the irreducible representation corresponding to \((\chi_1,\chi_2)\) is isomorphic to the one corresponding to \((\chi_2,\chi_1)\).

We have \(\frac{1}{2}(q-1)(q-2)\) irreducible representations of degree \(q+1\).

All these representations are pairwise non-isomorphic.

We saw that there are \(\frac{(q^2-q)}{2}\) irreducible representations of degree \(q-1\) remaining.

These are all the representations of \(GL_2(\mathbb{F}_q)\) as can be checked from the following considerations.

We know that the order of \(GL_2(\mathbb{F}_q)\) is \((q^2-1)(q^2-q)\).

Recall that for a group of order \(n\) whose irreducible representations are \(\pi_1,\ldots, \pi_r\) of degrees \(d_1,\ldots,d_r\) respectively,

\begin{equation*}

n=d_1^2+\cdots + d_r^2.

\end{equation*}

The sum of squares of degrees of the representations that we have constructed so far is \((q^2-1)(q^2-q)\).