Denote $G = GL(2, q) = GL_2(\mathbb{F}_q)$, $B$ its Borel subgroup of upper triangular matrices, $T$ its splitting torus of diagonal matrices. The object I am interested in is $Ind_B^G\rho$, where $\rho$ is a $(q - 1)$-dimensional irreducible representation of $B$.

$$[G: B] = \frac{q(q + 1)(q - 1)^2}{q(q - 1)^2} = q + 1 \Longrightarrow \dim Ind_B^G\rho = (q + 1)(q - 1).$$

We know that $B$ have $(q - 1)^2$ many $1$-dimensional irreducible representations (determinant composed by two characters of $\mathbb{F}_q^*$), and $(q - 1)$ many $(q - 1)$-dimensional irreducible representations $\rho$. We can think about them as part of $Ind_T^G\tau$ ($\tau$ an irreducible representation of $T \cong \mathbb{F}_q^* \times \mathbb{F}_q^*$), which decomposes to the direct sum of two irreducible representations of dimension $1$ and $(q - 1)$.

While representations of $G$ induced from $1$-dimensional representations of $B$ give principal series representations of $G$, representations of $G$ induced from $(q - 1)$-dimensional representations of $B$ contains the cuspidal representations of $G$.

Question. How does $Ind_B^G\rho$ ($\dim \rho = q - 1$ irreducible) decompose into irreducible representations of $G$?


If $\chi$ is a nontrivial character of the unipotent radical $U$ of $B$ then $\text{Ind}_U^G(\chi)$ is the sum of all irreducible non one-dimensional representations of G with multiplicity one. See See Piatetski-Shapiro, Complex representations of GL(2,K) for finite fields K, Contemporary Mathematics, 1983 (MR0696772), page 2.Therefore $\text{Ind}_B^G(\rho)$ is the sum of all irreducible non one-dimensional representations, having a given central character, with multiplicity one.


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