Scopes reduction and Morita equivalence classes of blocks in finite classical groups. II. (Q1763738)

[For part I cf. the authors, ibid. 230, No. 2, 378-423 (2000; Zbl 0961.20036)]. Donovan has conjectured that, for any prime \(p\) and any finite \(p\)-group \(D\), the number of Morita equivalence classes of \(p\)-blocks of finite groups with defect group \(D\) is finite. In the paper under review, the authors are concerned with this conjecture for unipotent \(p\)-blocks of one of the classical groups \(\text{SL}_n(q)\), \(\text{SU}_n(q)\), \(\text{SO}_{2n+1}(q)\), \(\text{Sp}_{2n}(q)\) and \(\text{SO}^\pm_{2n}(q)\). For these groups and blocks, they obtain a reduction to the following situations: (1) \(\text{SU}_n(q)\), \(\text{SO}_{2n+1}(q)\), \(\text{Sp}_{2n}(q)\), \(\text{SO}^\pm_{2n}(q)\) for some \(n\) bounded in terms of \(|D|\); (2) \(\text{SO}^\pm_{2n}(q)\) for some \(n\) and some prime power \(q\equiv -1\pmod p\). Their proof is based on a new general criterion for Morita (and Puig) equivalences of blocks.