On Cuspidal Representations of General Linear Groups over Discrete Valuation Rings

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 $k\ge 2$ 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.

20 pages (revised)

References in corpus (1)

• On Representations of General Linear Groups over Principal Ideal Local Rings of Length Two

Cited by in corpus (11)

• On Representations of General Linear Groups over Principal Ideal Local Rings of Length Two
• On character values and decomposition of the Weil representation associated to a finite abelian group
• Branching Rules for Supercuspidal Representations of SL_2(k)
• Combinatorics of finite abelian groups and Weil representations
• Regular characters of groups of type A_n over discrete valuation rings
• Representations of automorphism groups of finite O-modules of rank two
• A multiplicity one theorem for groups of type $A_n$ over discrete valuation rings
• A characterization of groups of parahoric type
• On the cohomology of the Lubin-Tate curve of level 2 and the Lusztig theory over finite rings
• Extended Deligne-Lusztig varieties for general and special linear groups
• Flags and orbits of connected reductive groups over local rings