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

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*, .

## 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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## More information

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

## Catalogue record

Date deposited: 27 Nov 2009

Last modified: 19 Jul 2017 00:07

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## Contributors

Author:
Anne-Marie Aubert

Author:
Uri Onn

Author:
Amritanshu Prasad

Author:
Alexander Stasinski

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