DOCUMENTA MATHEMATICA, Extra Volume ICM II (1998), 121-128

Jeremy Rickard 

Title: The Abelian Defect Group Conjecture 

Let $G$ be a finite group and $k$ an algebraically closed field 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.