Vers une décomposition de Jordan des blocs des groupes réductifs finis

Let G be a reductive algebraic group over an algebraic closure of a prime field , defined over , with Frobenius endomorphism F. Let GF be the subgroup of rational points. Let ℓ be a prime number. Assume that ℓ is different from p. If (G*,F) is in duality with (G,F), then, by a theorem of M. Broué and J. Michel [M. Broué, J. Michel, Blocs et séries de Lusztig dans un groupe réductif fini, J. Reine Angew. Math. 395 (1989) 56–67], for any ℓ-bloc B of GF there exists a unique (G*)F-conjugacy class (s) of ℓ′-semi-simple elements such that some irreducible representation of B is in the rational Lusztigʹs series associated to (s). If s=1, B is said to be unipotent. If G is not connected, with identity component G○, define the “unipotent ℓ-blocs of GF” as the ℓ-blocks that cover some unipotent ℓ-bloc of (G○)F. Assuming ℓ is good for G, we construct from (G,F) and (s) some reductive algebraic group (G(s),F) defined over and a one-to-one map from the set of ℓ-blocks of GF with associated class (s) onto the set of “unipotent” ℓ-blocks of G(s)F such that, if b corresponds to B, then – there is a significant height preserving one-to-one map from the set of irreducible representations Irr(B) onto the set Irr(b), – the respective defect groups of b and B are isomorphic, the associated Brauer categories are equivalent. One do not assume that the center of G is connected, the class of s may be isolated in G*.