Crystal bases and simple modules for hecke algebras of type g(_{p, p, n})

We apply the crystal basis theory for Fock spaces over quantum affine algebras to the modular representations of the cyclotomic Hecke algebras of type G(p, p, n). This yields a 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 and Sagaki’s work [S. Naito & D. Sagaki, J. Algebra 251, (2002) 461-474] on Lakshmibai-Seshadri paths fixed by diagram automorphisms to derive explicit formulas for the number of simple modules over these Hecke algebras. These formulas 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)).