The Abelian Defect Group Conjecture

Let G be a nite group and k an algebraically closed eld of characteristic p > 0. If B is a block of the group algebra kG with defect group D, the Brauer correspondent of B is a block b of kN G (D). When D is abelian, the blocks B and b, although they are rarely isomorphic or even Morita equivalent, seem to be very closely related. For example, Alperin's Weight Conjecture predicts that they should have the same number of simple modules. Brou e's Abelian Defect Group Conjecture gives a more precise prediction of the relationship between B and b: their module categories should have equivalent derived categories. In this article we survey this conjecture, some of its consequences, and some of the recent progress that has been made in verifying it in special cases. 1 Notation and terminology Throughout this article, G will denote a nite group. We shall be dealing with the characteristic p representation theory of G, where p is a prime. We shall use three coeecient rings. The ring O will be a complete discrete valuation ring with residue eld k of characteristic p and eld of fractions K of characteristic zero. Since we shall not be concerned with rationality questions, we shall assume that these coeecient rings are alìlarge enough' in that they contain enough roots of unity.