The Characters of the Generalized Steinberg Representations of Finite General Linear Groups on the Regular Elliptic Set

Let k be a finite field, kn|k the degree n extension of k, and G = GLn(k) the general linear group with entries in k. This paper studies the “generalized Steinberg” (GS) representations of G and proves the equivalence of several different characterizations for this class of representations. As our main result we show that the union of the class of cuspidal and GS representations of G is in natural one-one correspondence with the set of Galois orbits of characters of k× n , the regular orbits of course corresponding to the cuspidal representations. Besides using Green’s character formulas to define GS representations, we characterize GS representations by associating to them idempotents in certain commuting algebras corresponding to parabolic inductions and by showing that GS representations are the sole components of these induced representations which are “generic” (have Whittaker vectors). Let k = Fq be a finite field of cardinality q, let k̄|km|k be, respectively, an algebraic closure of k and the degreem extension of k contained in k̄. Let φ : x 7→ x denote the Frobenius automorphism of k̄|k and of any subextension km|k. Let G = GLn(k) be the group of non-singular n × n matrices with entries in k. Also write Gm = GLm(k) for any m ≥ 1. In a signally important paper published in 1955 [GR] J. A. Green showed how to calculate formulas for the irreducible characters of G. In Green’s work appeared for the first time general character formulas for the cuspidal representations and for a family of “generalized Steinberg” (GS) representations. For their study of the “level zero” discrete series characters of unit groups of simple algebras over a p–adic field the authors need diverse characterizations of cuspidal and GS characters of finite general linear groups and to be able to pass between these different characterizations. In this paper we give these characterizations and prove their equivalence. It is fruitful to view the set of GS representations as a class of representations of G which contains the class of cuspidal representations as a subclass. Of course the cusp form property, the property of not being a component of IndU 1 for any unipotent radical U 6= (I) of a parabolic subgroup of G, clearly distinguishes the class of cuspidal representations from all other representations of G. However, other important properties which are usually associated to the class of cuspidal representations generalize to the union of the two classes of representations of G. Received by the editors May 26, 1997 and, in revised form, April 18, 1998 and June 26, 1998. 1991 Mathematics Subject Classification. Primary 22E50, 11T24.