Let $G$ be a finite group, let $k$ be a large enough field of characteristic $p>0$. Let $p\mid |G|$.


**Broué's abelian defect group conjecture** states the following:


Let $B$ be a block of $kG$ with
abelian defect group $D$ and let $b\in \text{Bl}(N\_G(D))$ be its Brauer correspondent. Then the derived categories $D^b (\text{mod}(B))$ and $D^b(\text{mod}(b))$ of bounded complexes of finitely generated modules over $B$ and $b$ are equivalent as triangulated categories.


**Alperin's weight conjecture (blockwise version)** states the following:


Let $B$ be a block of $kG$. Then the number of irreducible Brauer characters belonging to $B$ equals the number of $G$-conjugacy classes of $p$-weights of $B$.



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> Is there a relationship between these two conjectures? For example: does one conjecture imply the other?
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The motivation for this question is: both Alperin's weight conjecture and Broué's abelian defect group conjecture can be expressed via trivial source modules (\*).


Thank you very much for the help.


(\*) Small footnote: in the case of Broué's abelian defect group conjecture this means that all modules occurring in the cochain complex realising such a derived equivalence are supposed to be $p$-permutation modules. Hence, this is more restrictive than the general conjecture.



Is there a relationship between Broué's abelian defect group conjecture and Alperin's weight conjecture?
Yes. Broué's conjecture implies the abelian defect case of Alperin's weight conjecture. This makes one think there might be a structural statement like Broué's conjecture, for non-abelian defect groups, that would imply Alperin's conjecture in general. Many attempts have been made, to little avail; Rouquier has conjectured that if the hyperfocal subgroup of the defect group is abelian then there is a derived equivalence as in Broué's conjecture, but that's about it.


