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We apply the crystal basis theory for Fock spaces over quantum affinealgebras to the modular representations of the cyclotomic Hecke algebras oftype $G(p,p,n)$. This yields a classification of simple modules over thesecyclotomic Hecke algebras in the non-separated case, generalizing our previouswork [J. Hu, J. Algebra 267 (2003) 7-20]. The separated case was completed in[J. Hu, J. Algebra 274 (2004) 446--490]. Furthermore, we use Naito--Sagaki'swork [S. Naito & D. Sagaki, J. Algebra 251 (2002) 461--474] onLakshmibai--Seshadri paths fixed by diagram automorphisms to derive explicitformulas for the number of simple modules over these Hecke algebras. Theseformulas generalize earlier results of [M. Geck, Represent. Theory 4 (2000)370-397] on the Hecke algebras of type $D_n$ (i.e., of type $G(2,2,n)$).

Crystal bases and simple modules for Hecke algebras of type $G(p,p,n)$