Blocks with a single automorphism orbit of irreducible Brauer characters

Abstract

Fix a prime $p$. Let $G⊴\tilde{G}$ be finite groups such that $|\tilde{G}:G|$ is a power of $p$, and let $\tilde{b}$ be a block of $\tilde{G}$ with abelian defect groups covering a block $b$ of $G$. We prove that $\tilde{b}$ is inertial whenever $\IBr(b)$ is a single $\Aut(G)_b$-orbit. This verifies the abelian-defect case of the Kessar--Linckelmann conjecture. As a key ingredient, we further show that a block of a finite quasisimple group with a single automorphism orbit of irreducible Brauer characters has abelian defect groups. As applications, we prove the blockwise Alperin weight conjecture for blocks with exactly one irreducible Brauer character and establish Puig's nilpotency conjecture.