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We define a new notion of cuspidality for representations of $\GL_n$ over afinite quotient $\Oh_k$ of the ring of integers $\Oh$ of a non-Archimedeanlocal field $F$ using geometric and infinitesimal induction functors, whichinvolve automorphism groups $G_\lambda$ of torsion $\Oh$\nobreakdash-modules.When $n$ is a prime, we show that this notion of cuspidality is equivalent tostrong cuspidality, which arises in the construction of supercuspidalrepresentations of $\GL_n(F)$. We show that strongly cuspidal representationsshare many features of cuspidal representations of finite general lineargroups. In the function field case, we show that the construction of therepresentations of $\GL_n(\Oh_k)$ for $k\geq 2$ for all $n$ is equivalent tothe construction of the representations of all the groups $G_\lambda$. Afunctional equation for zeta functions for representations of $\GL_n(\Oh_k)$ isestablished for representations which are not contained in an infinitesimallyinduced representation. All the cuspidal representations for $\GL_4(\Oh_2)$ areconstructed. Not all these representations are strongly cuspidal.

On Cuspidal Representations of General Linear Groups over Discrete Valuation Rings