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In the 1980's, the magic properties of the cohomology of elementary abeliangroups as modules over the Steenrod algebra initiated a long lastinginteraction between topology and modular representation theory in naturalcharacteristic. The Adams-Gunawardena-Miller theorem in particular, showed thattheir decomposition is governed by the modular representations of thesemi-groups of square matrices. Applying Lannes' T functor on the summands L P:= Hom Mn(Fp) (P, H * (F p) n) defines an intriguing construction inrepresentation theory. We show that T(L P) $\sim$ = L P $\oplus$ H * V 1$\otimes$ L $\delta$(P) , defining a functor $\delta$ from F p [M n (Fp)]-projectives to F p [M n--1 (F p)]-projectives. We relate this new functor$\delta$ to classical constructions in the representation theory of the generallinear groups.

Lannes' t functor on injective unstable modules and harish-chandra restriction