Questions about cuspidal representations of $\operatorname{GL}_2(\mathbb{F}_q)$. All representations of $\operatorname{GL}_2(\mathbb{F}_q)$ are classified in the book. They are principal series representations, complementary series representations, 1-dimensional representations. They form all representations of $\operatorname{GL}_2(\mathbb{F}_q)$.
In the notes, some representations of $\operatorname{GL}_2(\mathbb{F}_q)$ are called cuspidal representations, and some are called parabolic induced representations.
Are parabolic induced representations the same as principal series representations and cuspidal representations the same as complementary series representations? Thank you very much.
 A: $\newcommand{\F}{\mathbb{F}}$$\newcommand{\Ind}{\text{Ind}}$Terminology isn't always completely agreed upon. This is how I remember the terminology for $GL_2(\F_q)$: 
First: The cuspidal representations are exactly the irreducible represenations that do not contain the trivial character of
$$
N = \pmatrix{1 & * \\ 0 & 1}.
$$
The one dimensional representations/characters are those of this form
$$
g \mapsto \xi(\det(g))
$$
where $\xi$ is a character of $\F_q^{\times}$. 
The principal series representations: Let 
$$
\lambda \pmatrix{a & b \\ 0 & c} = \lambda_1(a)\lambda_2(b).
$$
be a character of 
$$
B = \pmatrix{* & * \\ 0 & *}.
$$
where $\lambda_i$ are characters of $\F_q^{\times}$. Then
$$
V_{\lambda_1, \lambda_2} = \Ind_B^G(\lambda)
$$
is an irreducible representation of $G$ exactly when $\lambda_1 \neq \lambda_2$. These representations have been constructed using parabolic induction and fall into that category. They all contain the trivial character of $N$ and are therefore not cuspidal. There are $\frac{1}{2}(q^2 +q) -1$ of these. Note in particular that the one dimensional representations show up when $\Ind_B^G(\lambda)$ is reducible. And so I think we say that the one-dimensional representations in the principal series.
Complementary series representations: These are the cuspidal representations and you can refer to your book for the construction. There are $\frac{1}{2}q(q-1)$ of these
