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

Vincent Franjou (1) , Nguyen Dang Ho Hai , Lionel Schwartz (2)

1 Topologie, géométrie et algèbre
2 LAGA - Laboratoire Analyse, Géométrie et Applications

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 LP := Hom_Mn(n,Fp)(P,H*(Z/p)^n) defines an intriguing construction in representation theory. We show that: T(LP) = LP ⊕ H*(Z/p) ⊗ Lδ(P), for a functor δ from Fp[M(n,Fp)]-projectives to Fp[M(n−1,Fp)]-projectives. We relate this new functor δ to classical constructions in the representation theory of the general linear groups.

Domaines: Théorie des représentations [math.RT]; K-théorie et homologie [math.KT]; Topologie algébrique [math.AT]

Vincent Franjou, Nguyen Dang Ho Hai, Lionel Schwartz. Lannes’ T functor on injective unstable modules and Harish-Chandra restriction. International Mathematics Research Notices, 2017, ⟨10.1093/imrn/rnw298⟩. ⟨hal-02387693⟩