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On cuspidal representations of general linear groups over discrete valuation rings

On cuspidal representations of general linear groups over discrete valuation rings
On cuspidal representations of general linear groups over discrete valuation rings
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
0021-2172
21pp
Aubert, Anne-Marie
f1ab184a-28bd-49a7-bec5-d0abcec2fed0
Onn, Uri
47e1ca18-b0f2-4cd9-b857-2f6cb1a5f071
Prasad, Amritanshu
8608c953-c8f7-4ee1-ac42-b987f806d1f2
Stasinski, Alexander
94bd8be7-4b4f-4e22-875b-3628d8c2ca19
Aubert, Anne-Marie
f1ab184a-28bd-49a7-bec5-d0abcec2fed0
Onn, Uri
47e1ca18-b0f2-4cd9-b857-2f6cb1a5f071
Prasad, Amritanshu
8608c953-c8f7-4ee1-ac42-b987f806d1f2
Stasinski, Alexander
94bd8be7-4b4f-4e22-875b-3628d8c2ca19

Aubert, Anne-Marie, Onn, Uri, Prasad, Amritanshu and Stasinski, Alexander (2009) On cuspidal representations of general linear groups over discrete valuation rings. Israel Journal of Mathematics, 21pp.

Record type: Article

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

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Submitted date: 2009

Identifiers

Local EPrints ID: 69674
URI: https://eprints.soton.ac.uk/id/eprint/69674
ISSN: 0021-2172
PURE UUID: 36bd9513-4b9e-48a7-a2c9-706cbb84d303

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Date deposited: 27 Nov 2009
Last modified: 19 Jul 2017 00:07

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