Parabolically induced representations of $GL_n(O_l)$

Let $\mathcal O$ be the ring of integers of a non-Archimedean local field with uniformizer $\varpi$, and let $\mathcal O_\ell=\mathcal O/\varpi^\ell\mathcal O$ for $\ell\geq1$. In this article, we study representations of $\mathrm{GL}_n(\mathcal O_\ell)$ parabolically induced from characters of the