Question

Fix a finite extension $F$ of $\mathbb{Q}_p$. Consider its ring of integers $\mathfrak{o}$ with maximal ideal $\mathfrak{p}$. Set $R_n = \mathfrak{o}/\mathfrak{p}^n$. Let $\mathrm{B}$ be the upper triangular matrices. Consider a character $\mu : \mathrm{B}(R_n)\rightarrow \mathbb{C}^\times$. When is the induced representation $ \mathrm{Ind}_{\mathrm{B}(R_n)}^{\mathrm{GL}_2(R_n)} \mu$ irreducible?

Answer 1

A sufficient criterion for irreducibility is given, for example, in Theorem 4.6 in Hill: Semisimple and cuspidal characters of $\mathrm{GL}_n(\mathcal{O})$. Hill's result is more general, and holds for certain representations of $\mathrm{GL}_n(\mathcal{O})$, for $n\geq 2$. For $\mathrm{GL}_2(\mathcal{O})$ it says the following. Let $T$ be the diagonal torus, so that $T(\mathcal{O}_r)\cong\mathcal{O}_r^{\times}\times\mathcal{O}_r^{\times}$. Let $\theta=\theta_1\theta_2$ be a character of $T(\mathcal{O}_r)$, where $\theta_1$ and $\theta_2$ are characters of $\mathcal{O}_r^\times$. Suppose that the restriction of $\theta_1$ to $1+\mathfrak{p}^{r-1}$ is non-trivial, and that the restriction of $\theta_1$ to $1+\mathfrak{p}^{r-1}$ is not equal to that of $\theta_2$. Let $\tilde{\theta}$ denote the pull-back of $\theta$ to $B(\mathcal{O}_r)$. Then the representation $$\mathrm{Ind}^{\mathrm{GL}_2(\mathcal{O}_r)}_{B(\mathcal{O}_r)}\tilde{\theta}$$ is irreducible.

Answer 2

For PGL(2) I have the following reference :

Silberger, A.J. : PGL 2 over the p-adics : its representations, spherical functions, and Fourier analysis. Springer Lecture Notes in Mathematics 166. Berlin, Heidelberg, New York: Springer 1970

I don't have this book with me, but the basic idea is to apply Mackey's irreducibility criterion. For this you have to determine the double cosets of $G={\rm GL}_2 (R_u )$ mod $B=B(R_u )$. For $n=1$, just use the Bruhat decomposition. To get a set of representatives of the double quotient for $n>1$, you introduce the "Iwahori" subgroup $I$ of matrices that are upper triangular mod ${\mathfrak p}$. Then $G=I\cup IwI$, where $w$ is the standard Weyl group element. Next you have the following set of representatives of right (or left) $B$-cosets in $I$ : the lower triangular unipotent matrices with the coefficient varying over ${\mathfrak p}/{\mathfrak p}^{n}$.