Suppose $G$ is a finite group, and $l$ is a prime, with $l$ coprime to the order $|G|$. (Thus we have complete reducibility for $G$ representations.) Is there a straightforward condition on $l$ which ensures that every irreducible representation of $G$ is liftable to a characteristic zero representation? (For instance, does the fact that we assume $l$ coprime to $|G|$ suffice?)



When are all characteristic l representations liftable
Yes, $\operatorname{gcd}(l,|G|) = 1$ is sufficient. This is an easy consequence of Brauer's modular representation theory. See Serre's *Linear Representations of Finite Groups*, especially Chapter 18.


