What is Broué's Abelian Defect Group Conjecture?

This is not intended to be a comprehensive discussion of the conjecture. For more details see, for example, [Bro2], [Bro4], [Bro5], [KonZim], [Ric7]. Let p be a prime, let R be a complete discrete valuation ring of characteristic zero with residue field k of characteristic p, and suppose that R and k are `large enough' (i.e., contain enough roots of unity for the groups that we will consider).

Let G be a finite group, let A be a block algebra of the group algebra RG that has an abelian defect group D, and let B be the Brauer correspondent of A, a block algebra of RN_G(D), the group algebra of the normalizer of D in G.

Then Broué conjectured in the 1990 paper [Bro2] that

Broué's Abelian Defect Group Conjecture

The derived categories D^b(mod-A) and D^b(mod-B) of bounded complexes of finitely generated modules for A and B are equivalent as triangulated categories.

If this is true, it has many consequences. For example:

The corresponding blocks over k have equivalent derived categories.
The blocks have the same number of irreducible ordinary characters.
The blocks have the same number of irreducible Brauer characters.
The blocks have isomorphic centres.
The blocks have the same Hochschild cohomology algebras.
The blocks have the same algebraic K-theory.

The character-theoretic consequences are subsumed by the statement that there is a `perfect isometry' between the blocks. This means an isometry between the groups of virtual characters for the two blocks that preserves the subgroups spanned by characters of projective modules. An isometry amounts to a bijection `with signs' between the sets of irreducible ordinary characters; the other condition can be restated as an arithmetical condition about character values.