Blocks with only one irreducible Brauer character orbit

Abstract

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 Brou\'e'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.