Blocks with only one irreducible Brauer character orbit

In this paper, we confirm the Kessar-Linckelmann conjecture for blocks with abelian defect groups.

More generally, we prove that if a covered block with abelian defect groups has its irreducible Brauer characters forming a single orbit under the block stabilizer, then the covering block is inertial; we term these blocks with a single irreducible Brauer character orbit.

Consequently, we establish Broué's abelian defect conjecture for such blocks.

As a byproduct, we show that blocks of finite quasisimple groups with a single irreducible Brauer character orbit necessarily have abelian defect groups.

As applications, we verify the blockwise Alperin weight conjecture for blocks with a unique irreducible Brauer character and prove Puig's long-standing conjecture in full generality.

All results rely essentially on the Classification of finite simple groups.