Morita equivalence for blocks of Hecke algebras of type \(B\) (Q1305066)

The author continues his work [from J. Algebra 194, No. 1, 201-223 (1997; Zbl 0883.20009) and Manuscr. Math. 91, No. 1, 121-144 (1996; Zbl 0896.20009)] and shows that given a pair \(w=(w_1,w_2)\) of nonnegative integers, a complete discrete valuation ring \(R\) and parameters \(q\), \(Q\) in \(R\), there is only a finite number of Morita equivalence classes of blocks of Hecke algebras of type \(B\) over \(R\) with parameters \(q\), \(Q\) whose weight tuple is \(w\). Let \(F\) denote the residue class field of \(R\). The notion of a combinatorial \((w:k)\)-pair is introduced which is then used to define \((w_i:k_i)\)-pairs of blocks \(B\), \(\overline B\) of Hecke algebras \(H_F(W_n)\) and \(H_F(W_{n-k_i})\) of type \(B\) over \(F\), respectively. A bijection between the simple modules in \(B\) and \(\overline B\) is set up, and from this it is deduced that the blocks \(B\) and \(\overline B\) have the same decomposition matrices and hence also the same Cartan matrices. It is then shown that the blocks \(B\) and \(\overline B\) are Morita equivalent by setting up a pair of functors \(G\), \(G'\) between mod-\(B\) and mod-\(\overline B\). As a consequence the main result mentioned in the beginning is proved.