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In the 1980’s, the magic properties of the cohomology of elementary abelian groups as modules over the Steenrod algebra initiated a long lasting interaction between topology and modular representation theory in natural characteristic. The Adams-Gunawardena-Miller theorem in particular, showed that their decomposition is governed by the modular representations of the semi-groups of square matrices. Applying Lannes’ T functor on the summands   $L_P:=\mathrm{Hom}_{\mathrm {M}_n({\mathbb F}_p)}(P,H^*(Z/p (\mathbb{Z} / p))^n)$   defines an intriguing construction in representation theory. We show that   $\mathrm{T} (L_P) \cong L_P \oplus H^*(\mathbb{Z} / p)\,{\otimes}\, L_{\delta (P)}$  , for a functor   $\delta$   from   $\mathbb F_p[\mathrm {M}_n(\mathbb F_p)]$  -projectives to   $\mathbb F_p[\mathrm {M}_{n-1}(\mathbb F_p)]$  -projectives. We relate this new functor   $\delta$   to classical constructions in the representation theory of the general linear groups.

Lannes’ T functor on injective unstable modules and Harish-Chandra restriction