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?



Parabolic induction for GL(2,Z/pn)
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.


