Prime power degree representations of quasi-simple groups

Abstract.

We classify the irreducible complex characters of prime power degree of the finite quasi-simple groups different from the alternating groups and their covering groups. A well-known example of such characters is given by the Steinberg characters of finite groups of Lie type. It turns out that, apart from sporadic examples coming from numerical coincidences, like Fermat and Mersenne primes, this is the generic case. The proof proceeds via the classification of finite simple groups and makes essential use of Deligne-Lusztig theory.