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The characters of the generalized Steinberg representations of finite general linear groups on the regular elliptic set

Authors: Allan J. Silberger and Ernst-Wilhelm Zink
Journal: Trans. Amer. Math. Soc. 352 (2000), 3339-3356
MSC (1991): Primary 22E50, 11T24
Published electronically: March 24, 2000
MathSciNet review: 1650042
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Abstract: Let $k$ be a finite field, $k_{n}\vert k$ the degree $n$ extension of $k$, and $G=\operatorname{GL}_{n}(k)$ 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 $k_{n}^{\times }$, 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).

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Additional Information

Allan J. Silberger
Affiliation: Department of Mathematics, Cleveland State University, Cleveland, Ohio 44115

Ernst-Wilhelm Zink
Affiliation: Humboldt-Universität, FB Reine Mathematik, Unter den Linden 6, 10099 Berlin, Germany

Keywords: Reductive group, general linear group, finite field, character, unitary representation, Steinberg representation, Whittaker vector, generic representation
Received by editor(s): May 26, 1997
Received by editor(s) in revised form: April 18, 1998, and June 26, 1998
Published electronically: March 24, 2000
Article copyright: © Copyright 2000 American Mathematical Society