Towards a Jordan decomposition of blocks of finite reductive groups

\input amssym.def \input amssym.tex Let $G$ be a connected algebraic reductive group over an algebraic closure of a prime field ${\Bbb F}_p$, defined over ${\Bbb F}_q$ thanks to a Frobenius $F$. Let $\ell$ be a prime different from $p$. Let $B$ be an $\ell$-block of the subgroup of rational points $G^F$. Under mild restrictions on $\ell$, we show the existence of an algebraic reductive group $H$ defined over ${\Bbb F}_q$ {\it via} a Frobenius $F$, and of a unipotent $\ell$-block $b$ of $H^F$ such that : the respective defect groups of $b$ and $B$ are isomorphic, the associated Brauer categories are isomorphic and there is a height preserving one-to-one map from the set of irreducible representations of $b$ onto the set of irreducible representations of $B$. \end