  Let G be a finite non-abelian simple group and let p be a prime. We classify
all pairs (G,p) such that the sum of the complex irreducible character degrees
of G is greater than the index of a Sylow p-subgroup of G. Our classification
includes all groups of Lie type in defining characteristic p (because every
Gelfand-Graev character of G is multiplicity free and has degree equal to the
above index), and a handful of well-described examples.
