Quasi-isolated blocks and Brauer's height zero conjecture

Abstract

This paper has two main results. Firstly, we complete the parametrisation of all p-blocks of finite quasi-simple groups by finding the so-called quasi-isolated blocks of exceptional groups of Lie type for bad primes. This relies on the explicit decomposition of Lusztig induction from suitable Levi subgroups. Our second major result is the proof of one direction of Brauer's long-standing height zero conjecture on blocks of finite groups, using the reduction by Berger and Kn\"orr to the quasi-simple situation. We also use our result on blocks to verify a conjecture of Malle and Navarro on nilpotent blocks for all quasi-simple groups. In the supplement we treat a configuration for the $5$-blocks of the finite simple groups $E_8(q)$ that was inadvertently omitted in the first version and clarify and correct the meaning of some entries in Tables 3 and 4. The main theorems of the original version are not affected.