Fix a finite extension $F$ of $\mathbb{Q}_p$. Consider its ring of integers $\mathfrak{o}$ with maximal ideal $\mathfrak{p}$. Set $R_n = \mathfrak{o}/\mathfrak{p}^r$. 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?

Fix a finite extension $F$ of $\mathbb{Q}_p$. Consider its ring of integers $\mathfrak{o}$ with maximal ideal $\mathfrak{p}$. Set $R_n = \mathfrak{o}/\mathfrak{p}^r$. 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?