Splendid Equivalences: Derived Categories and Permutation Modules

We introduce the notion of a splendid equivalence between blocks of finite groups, which is a strengthening of the notion of an equivalence of derived categories. Such an equivalence is always induced by a two-sided tilting complex whose terms are summands of permutation modules, a feature that is compatible with the belief that these complexes should be related to the l-adic cohomology of Deligne–Lusztig varities in the case of reductive groups over finite fields. We show that a splendid equivalence has good consequences over and above those that follow from a derived equivalence. In particular, it gives, for principal blocks, a structural explanation of Broué's concept of an isotypy (a compatible family of perfect isometries). We give several examples, including examples of Morita equivalences that are splendid in an unexpectedly non-trivial way.