A height-zero type result for blocks of solvable groups

Abstract

Let $B$ be a $p$-block of a finite group $G$ with defect group $D$. The more difficult direction of the recently proven height zero conjecture says that $D$ is abelian if every character in Irr$(B)$ has height zero. We consider a smaller set than Irr$(B)$. In particular, if $\varphi \in {\rm IBr}_p(B)$, we let Irr$(\varphi)$ be the set of characters $\chi \in {\rm Irr}(G)$ such that $\varphi$ is a constituent of $\chi^o$. Now suppose $G$ is solvable and $\varphi$ is a height zero Brauer character in some block $B$ of $G$ with defect group $D$. Here we show that if every character in Irr$(\varphi)$ has height zero, then the defect group $D$ of the block containing $\varphi$ is abelian for $p \geq 5$ and almost abelian for $p = 2$ or $3$. This has a nice consequence for primitive characters of $p$-complements in solvable groups.