This paper is a continuation and a completion of the work of the first and the third author on the Jordan decomposition. We extend the Jordan decomposition of blocks: we show that blocks of finite groups of Lie type in nondescribing characteristic are Morita equivalent to blocks of subgroups associated to isolated elements of the dual group — this is the modular version of a fundamental result of Lusztig, and the best approximation of the character-theoretic Jordan decomposition that can be obtained via Deligne-Lusztig varieties. The key new result is the invariance of the part of the cohomology in a given modular series of Deligne-Lusztig varieties associated to a given Levi subgroup, under certain variations of parabolic subgroups.\looseness=-1
We also bring in local block theory methods: we show that the equivalence arises from a splendid Rickard equivalence. Even in the setting of the original work of the first and the third author, the finer homotopy equivalence was unknown. As a consequence, the equivalences preserve defect groups and categories of subpairs. We finally determine when Deligne-Lusztig induced representations of tori generate the derived category of representations. An additional new feature is an extension of the results to disconnected reductive algebraic groups, which is required to handle local subgroups.