On Cuspidal Representations of General Linear Groups over Discrete Valuation Rings

TL;DR

This paper introduces a new notion of cuspidality for representations of general linear groups over finite quotients of local rings, linking it to supercuspidal representations and constructing all cuspidal representations for specific cases.

Contribution

It defines a novel cuspidality concept using geometric and infinitesimal induction, and establishes equivalences and constructions for representations over finite quotients.

Findings

New notion of cuspidality for $ ext{GL}_n$ over finite quotients.

Equivalence of cuspidality and strong cuspidality when $n$ is prime.

Complete construction of all cuspidal representations for $ ext{GL}_4( ext{O}_2)$.

Abstract

We define a new notion of cuspidality for representations of $\GL_n$ over a finite quotient $\Oh_k$ of the ring of integers $\Oh$ of a non-Archimedean local field $F$ using geometric and infinitesimal induction functors, which involve automorphism groups $G_\lambda$ of torsion $\Oh$\nobreakdash-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 $\GL_n(F)$. 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 $\GL_n(\Oh_k)$ 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 $\GL_n(\Oh_k)$ is established for representations which are not contained in an infinitesimally induced representation. All the cuspidal representations for $\GL_4(\Oh_2)$ are constructed. Not all these representations are strongly cuspidal.