TY - JOUR

T1 - On cuspidal representations of general linear groups over discrete valuation rings

AU - Aubert, Anne Marie

AU - Onn, Uri

AU - Prasad, Amritanshu

AU - Stasinski, Alexander

N1 - Funding Information:
∗ Supported by the Israel Science Foundation, ISF grant no. 555104, by the Edmund Landau Minerva Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany). ∗∗ Supported by XI plan project 11-R&D-IMS-5.01-0500. † Supported at various times by EPSRC Grants GR/T21714/01 and EP/C527402. Received December 24,2007 and in revised form July 11, 2008

PY - 2010/1/1

Y1 - 2010/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=77949969407&partnerID=8YFLogxK

U2 - 10.1007/s11856-010-0016-y

DO - 10.1007/s11856-010-0016-y

M3 - Article

AN - SCOPUS:77949969407

SN - 0021-2172

VL - 175

SP - 391

EP - 420

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

IS - 1

ER -