Splendid Derived Equivalences for Blocks of Finite Groups

Abstract

A central issue in finite group modular representation theory is the relationship between the p-local structure and the p-modular representation theory of a given finite group. In [5], Broué poses some startling conjectures. For example, he conjectures that if e is a p-block of a finite group G with abelian defect group D and if f is the Brauer correspondent block of e of the normalizer, NG(D), of D then e and f have equivalent derived categories over a complete discrete valuation ring with residue field of characteristic p. Some evidence for this conjecture has been obtained using an important Morita analog for derived categories of Rickard [11]. This result states that the existence of a tilting complex is a necessary and sufficient condition for the equivalence of two derived categories. In [5], Broué also defines an equivalence on the character level between p-blocks e and f of finite groups G and H that he calls a ‘perfect isometry’ and he demonstrates that it is a consequence of a derived category equivalence between e and f. In [5], Broué also poses a corresponding perfect isometry conjecture between a p-block e of a finite group G with an abelian defect group D and its Brauer correspondent p-block f of NG(D) and presents several examples of this phenomena. Subsequent research has provided much more evidence for this character-level conjecture. In many known examples of a perfect isometry between p-blocks e, f of finite groups G, H there are also perfect isometries between p-blocks of p-local subgroups corresponding to e and f and these isometries are compatible in a precise sense. In [5], Broué calls such a family of compatible perfect isometries an ‘isotypy’. In [11], Rickard addresses the analogous question of defining a p-locally compatible family of derived equivalences. In this important paper, he defines a ‘splendid tilting complex’ for p-blocks e and f of finite groups G and H with a common p-subgroup P. Then he demonstrates that if X is such a splendid tilting complex, if P is a Sylow p-subgroup of G and H and if G and H have the same ‘p-local structure’, then p-local splendid tilting complexes are obtained from X via the Brauer functor and ‘lifting’. Consequently, in this situation, we obtain an isotypy when e and f are the principal blocks of G and H. Linckelmann [9] and Puig [10] have also obtained important results in this area. In this paper, we refine the methods and program of [11] to obtain variants of some of the results of [11] that have wider applicability. Indeed, suppose that the blocks e and f of G and H have a common defect group D. Suppose also that X is a splendid tilting complex for e and f and that the p-local structure of (say) H with respect to D is contained in that of G, then the Brauer functor, lifting and ‘cutting’ by block indempotents applied to X yield local block tilting complexes and consequently an isotypy on the character level. Since the p-local structure containment hypothesis is satisfied, for example, when H is a subgroup of G (as is the case in Broué's conjectures) our results extend the applicability of these ideas and methods.