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ON CUSPIDAL REPRESENTATIONS OF GENERAL LINEAR GROUPS OVER DICRETE VALUATION RINGS
 

Summary: ON CUSPIDAL REPRESENTATIONS OF GENERAL LINEAR GROUPS
OVER DICRETE VALUATION RINGS
ANNE-MARIE AUBERT, URI ONN , AMRITANSHU PRASAD,
AND WITH AN APPENDIX BY ALEXANDER STASINSKI
Abstract. 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 G 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 theory of GLn(ok) for k 2 for all n is equivalent to the construction
of the representations of all the groups G. A functional equation for zeta functions for
representations of GLn(ok) is established for representations which are not contained in
an infinitesimally induced representation. In the appendix, all cuspidal representations for
GL4(o2) are constructed. Not all these representations are strongly cuspidal.
1. Introduction
The irreducible characters of GLn(Fq) were computed by J. A. Green in 1955 [Gre55].
In Green's work, parabolic induction was used to construct many irreducible characters of
GLn(Fq) from irreducible characters of smaller general linear groups over Fq. The repre-

  

Source: Aubert, Anne-Marie - Institut de Mathématiques de Jussieu

 

Collections: Mathematics