A characterisation of nilpotent blocks

Let B be a p-block of a finite group, and set m = ∑χ(1)², the sum taken over all height zero characters of B. Motivated by a result of M. Isaacs characterising p-nilpotent finite groups in terms of character degrees, we show that B is nilpotent if and only if the exact power of p dividing m is equal to the p-part of |G : P|²|P : R|, where P is a defect group of B and where R is the focal subgroup of P with respect to a fusion system F of B on P. The proof involves the hyperfocal subalgebra D of a source algebra of B. We conjecture that all ordinary irreducible characters of D have degree prime to p if and only if the F-hyperfocal subgroup of P is abelian.