In this paper we study the rational representation
theory of the general linear group $G = \mbox{GL}_n(F)$
over an algebraically closed field $F$ of characteristic
$p$. Given $\alpha \in {\Bbb Z} / p{\Bbb Z}$, we define
functors $\mbox{Tr}^\alpha$ and $\mbox{Tr}_\alpha$,
which, roughly speaking, are given by tensoring
with the natural $G$-module $V$ and
its dual $V^*$ respectively, and then projecting onto
certain blocks determined by the residue $\alpha$.
In fact, these functors can be viewed as special cases of
Jantzen's translation functors.
We prove a number of fundamental properties about these
functors and also certain closely related functors
that arise in the modular representation theory of the symmetric group.
1991 Mathematics Subject Classification: 20G05, 20C05.