  Let G be a finite simple group of Lie type. In this paper we study characters
of G that vanish at the non-semisimple elements and whose degree is equal to
the order of a maximal unipotent subgroup of G. Such characters can be viewed
as a natural generalization of the Steinberg character. For groups G of small
rank we also determine the characters of this degree vanishing only at the
non-identity unipotent elements.
