On the Harris–Knörr correspondence in p-solvable groups

Abstract

Fix a prime p and let N be a normal subgroup of a finite p-solvable group G. Suppose that b is a p-block of N with abelian defect group Q and B is a p-block of G covering b. Let b⁎ be the Brauer correspondent of b in NN(Q) and let B⁎ be the unique p-block of NG(Q) that covers b⁎ and induces B. We show that there exist height preserving bijections Ψ of Irr(B) onto Irr(B⁎) and Ω of IBr(B) onto IBr(B⁎) such that the decomposition numbers dχφ and dΨ(χ)Ω(φ) are equal for all χ∈Irr(B) and all φ∈IBr(B).