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 \((q^2-1)(q^2-q)\).

   Log in to Reply