Modular Springer correspondence and decomposition matrices

Correspondance de Springer modulaire et matrices de décomposition

Daniel Juteau (1)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme

In 1976, Springer defined a correspondence making a link between the irreducible ordinary (characteristic zero) representations of a Weyl group and the geometry of the associated nilpotent variety. In this thesis, we define a modular Springer correspondence (in positive characteristic), and we show that the decomposition numbers of a Weyl group (for example the symmetric group) are particular cases of decomposition numbers for equivariant perverse sheaves on the nilpotent variety. We calculate explicitly the decomposition numbers associated to the regular and subregular classes, and to the minimal and trivial classes. We determine the correspondence explicitly in the case of the symmetric group, and show that James's row and column removal rule is a consequence of a smooth equivalence of nilpotent singularities obtained by Kraft and Procesi. The first chapter contains generalities about perverse sheaves with Z_l and F_l coefficients.

Mots clés: Springer correspondence; nilpotent singularities; modular representations; Weyl groups; perverse sheaves; intersection cohomology; integral cohomology

Domaines: Théorie des représentations [math.RT]; Géométrie algébrique [math.AG]

Daniel Juteau. Modular Springer correspondence and decomposition matrices. Representation Theory [math.RT]. Université Paris-Diderot - Paris VII, 2007. English. ⟨NNT : ⟩. ⟨tel-00355559⟩