Parabolically induced representations of $G{L}_n(O_l)$

Let $\mathcal{O}$ be the ring of integers of a non-Archimedean local field with uniformizer $\varpi$, and let ${\mathcal{O}}_{\mathrm{\ell }}=\mathcal{O}\mathrm{/}{\varpi }^{\mathrm{\ell }}\mathcal{O}$ for $\mathrm{\ell }\ge 1$. In this article, we study representations of ${\mathrm{G}\mathrm{L}}_n({\mathcal{O}}_{\mathrm{\ell }})$ parabolically induced from characters of the Borel subgroup ${\mathrm{B}}_n({\mathcal{O}}_{\mathrm{\ell }})$. For $n=3$, we establish a criterion for the irreducibility of these induced representations in terms of the relative conductors of the inducing characters. Motivated by this result, we formulate a conjecture characterizing the irreducibility of such representations for arbitrary $n$. We further show that when all inducing characters are equal, the resulting induced representation is not multiplicity-free, implying that the corresponding triple $({\mathrm{G}\mathrm{L}}_n({\mathcal{O}}_{\mathrm{\ell }}),{\mathrm{B}}_n({\mathcal{O}}_{\mathrm{\ell }}),\chi )$ is not a Gelfand pair. Finally, we study the irreducible decomposition of these representations for ${\mathrm{G}\mathrm{L}}_3({\mathcal{O}}_{\mathrm{\ell }})$ when two relative conductors attain their maximal values. Using the decomposition theory for ${\mathrm{G}\mathrm{L}}_2({\mathcal{O}}_{\mathrm{\ell }})$, we describe the irreducible constituents and determine their dimensions.