Correspondences of isomorphism types of irreducible modules in finite group block theory. (Q1945900)

Let \(k\) be an algebraically closed field of characteristic \(p>0\), let \(M\) be a normal subgroup of a finite group \(G\), and let \(B\) be a block of \(kM\) with defect group \(P\). The Harris-Knörr correspondence gives a bijection between blocks of \(kG\) covering \(B\) and blocks of \(kN_G(P)\) covering the Brauer correspondent \(B^*\) of \(B\), a block of \(kN_M(P)\) with defect group \(P\). Now suppose that \(M\) is \(p\)-solvable. Then a result by \textit{T. Okuyama} [Hokkaido Math. J. 10, 299-318 (1981; Zbl 0488.20013)] not mentioned in the paper under review gives a bijection between isomorphism classes of simple \(B\)-modules with vertex \(P\) and isomorphism classes of simple \(B^*\)-modules with vertex \(P\). The main result of the present paper shows that whenever \(A\) is a block of \(kG\) covering \(B\) and whenever \(A^*\) is the Harris-Knörr correspondent of \(A\) then there is a vertex-preserving bijection between the isomorphism classes of simple \(A\)-modules lying over a simple \(B\)-module \(S\) with vertex \(P\) and the isomorphism classes of simple \(A^*\)-modules lying over the simple \(B^*\)-module \(S^*\) corresponding to \(S\) in Okuyama's theorem. This result extends a theorem by \textit{A. Laradji} [J. Algebra 324, No. 4, 749-757 (2010; Zbl 1232.20015)].