On Translation Functors for General Linear and Symmetric Groups

In this paper we study the rational representation theory of the general linear group G = GLn(F) over an algebraically closed field F of characteristic p. Given α ∈ Z/pZ, we define functors Trα and Trα, 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 α. 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.