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

TL;DR

This paper explores the application of Lannes' T functor to injective unstable modules, revealing a new functor that connects modular representations of matrix semi-groups with classical general linear group representations.

Contribution

It introduces a novel functor $ ext{delta}$ derived from Lannes' T functor, linking injective unstable modules to classical representation theory of general linear groups.

Findings

Lannes' T functor decomposes injective modules into a direct sum involving a new functor.

The functor $ ext{delta}$ maps projectives over matrix semi-groups to lower-dimensional projectives.

Connections are established between this functor and classical representations of general linear groups.

Abstract

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 := Hom Mn(Fp) (P, H * (F p) n) defines an intriguing construction in representation 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 (F p)]-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 general linear groups.