# Automorphic forms and representations of $$GL_2$$ — Forum

## One thought on “Automorphic forms and representations of $$GL_2$$ — Forum”

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

1. When $$\chi_1\neq \chi_2$$, there is a unique irreducible representation of $$GL_2(\mathbb{F}_q)$$ of degree $$q+1$$ corresponding to $$\chi$$;
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 $$q+1$$.
2. When $$\chi_1=\chi_2$$, there are two irreducible representations of $$GL_2(\mathbb{F}_q)$$ corresponding to $$\chi$$, one of degree $$1$$ and the other of degree $$q$$.
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 $$(q^2-1)(q^2-q)$$.