On cuspidal representations of general linear groups over discrete valuation rings (Q5962258)

Let \(F\) be a non-Archimedean local field with ring of integers \(\mathfrak o\). Let \(\mathfrak p \) be the maximal ideal in \(\mathfrak o\), and \({\mathfrak o}_k={\mathfrak o}/{\mathfrak p}^k\) for \(k\geq 1\). The authors define a new notion of cuspidality for representations of \(\text{GL}_n\) over \({\mathfrak o}_k\) using geometric and infinitesimal induction functors, which involve automorphism groups \(G_\lambda\) of torsion \(\mathfrak o\)-modules. When \(n\) is a prime this notion of cuspidality is shown to be equivalent to strong cuspidality, which arises in the construction of supercuspidal representations of \(\text{GL}_n(F) \). Strongly cuspidal representations share many features of cuspidal representations of finite general linear groups. In the function field case, the construction of the representations of \(\text{GL}_n({\mathfrak o}_k)\) for \(k\geq 2\) for all \(n\) is equivalent to the construction of the representations of all the groups \(G_\lambda\). A functional equation for zeta functions for representations of \(\text{GL}_n({\mathfrak o}_k)\) is established for representations which are not contained in an infinitesimally induced representation. All cuspidal representations of \(\text{GL}_4({\mathfrak o}_2)\) are constructed. Not all of them are strongly cuspidal.