On Cuspidal Representations of General Linear Groups over Discrete Valuation Rings

We define a new notion of cuspidality for representations of GLn over a finite quotient ok of the ring of integers o of a non-Archimedean local field F using geometric and infinitesimal induction functors, which involve automorphism groups Gλ of torsion o-modules. When n is a prime, we show that this notion of cuspidality is equivalent to strong cuspidality, which arises in the construction of supercuspidal representations of GLn(F ). We show that strongly cuspidal representations share many features of cuspidal representations of finite general linear groups. In the function field case, we show that the construction of the representations of GLn(ok) for k ≥ 2 for all n is equivalent to the construction of the representations of all the groups Gλ. A functional equation for zeta functions for representations of GLn(ok) is established for representations which are not contained in an infinitesimally induced representation. All the cuspidal representations for GL4(o2) are constructed. Not all these representations are strongly cuspidal.