On quasi and weak Steinberg characters of general linear groups

Abstract Let G be a finite group and r be a prime divisor of the order of G . An irreducible character of G is said to be quasi r -Steinberg if it is non-zero on every r -regular element of G . A quasi r -Steinberg character of degree $\displaystyle |Syl_r(G)|$ is said to be weak r -Steinberg if it vanishes on the r -singular elements of $G.$ In this article, we classify the quasi r -Steinberg cuspidal characters of the general linear group $GL(n,q).$ Then we characterize the quasi r -Steinberg characters of $GL(2,q)$ and $GL(3,q).$ Finally, we obtain a classification of the weak r -Steinberg characters of $GL(n,q).$