Navarro vertices and lifts in solvable groups

Abstract

Let $Q$ be a $p$-subgroup of a finite $p$-solvable group $G$, where $p$ is a prime, and suppose that $\delta$ is a linear character of $Q$ with the property that $\delta (x)=\delta (y)$ whenever $x,y\in Q$ are conjugate in $G$. In this situation, we show that restriction to $p$-regular elements defines a canonical bijection from the set of those irreducible ordinary characters of $G$ with Navarro vertex $(Q,\delta )$ onto the set of irreducible Brauer characters of $G$ with Green vertex $Q$. Also, we use this correspondence to examine the behavior of lifts of Brauer characters with respect to normal subgroups.