2 7 Ju n 20 05 Crystal bases and simple modules for Hecke algebras of type G ( p , p , n ) ∗

We apply the crystal bases theory of Fock spaces over quantum affine algebras to the modular representations of the cyclotomic Hecke algebras of type G(p, p, n). This yields classification of simple modules over these cyclotomic Hecke algebras in the non-separated case, generalizing our previous work [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’s work [S. Naito & D. Sagaki, J. Algebra 251 (2002) 461–474] on Lakshmibai-Seshadri paths fixed by diagram automorphisms to derive explicit formula for the number of simple modules over these cyclotomic Hecke algebras. Our formula generalizes, in the special case where p = 2, earlier results of [M. Geck, Represent. Theory 4 (2000) 370-397] on the Hecke algebras of type Dn. ∗Keyword: Cyclotomic Hecke algebra, Fock spaces, crystal bases, Kleshchev multipartitions, Lakshmibai-Seshadri paths. ∗Research supported by National Natural Science Foundation of China (Project 10401005).