Cuspidal modules as summands of a Gelfand-Graev module

Tulunay, I.

Abstract

Let G=GL_n(q) be the general linear group over a finite field F_q with q elements. We call a Gelfand-Graev module to be the module which affords the Gelfand-Graev character. It is known that every cuspidal module of G is isomorphic to a (unique) direct summand of a Gelfand-Graev module. In this article, we investigate a certain endomorphism so that each irreducible cuspidal module is contained in a certain eigenspace corresponding to the cuspidal character. Furthermore, we determine the eigenvalue of that endomorphism by using character theory of finite general linear group.

Keywords: Group Representation Theory Linear algebraic groups over finite fields Representation theory of finite groups of Lie type.

AMS Subject Classification: Primary 20C33,; secondary 20G05, 20G40.