ON TRANSLATION FUNCTORS FOR GENERAL LINEAR AND SYMMETRIC GROUPS

Abstract

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.