Deligne–Lusztig characters of the finite general linear groups and eigenvectors of Lannes' T-functor

Abstract

The eigenvalue conjecture of Lionel Schwartz states that the induced action of Lannes' T-functor on the Grothendieck ring of reduced injective unstable modules over the mod. p Steenrod algebra is diagonalizable over Q with eigenvalues the powers of p. While some proofs have been given for the conjecture, none of them could shed light on the problem of determining the eigenvectors of T.The aim of this paper is to compute, at the same time, eigenvalues and eigenvectors of T by making use of the Deligne–Lusztig characters of the finite general linear groups. Based on this computation, some remarks and conjectures about the multiplicative structure of the Grothendieck ring and the Poincaré series of eigenvectors will be also given.