Representation theory of finite algebraic groups

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I.We studied complex raflection group G(r, 1, n) and associated cyclotomic Hecke algebras (Ariki-Koike algebra). Inparticular, by making use of the Schur-Weyl reciprocity proved in the paper (Sakamoto-Shoji), we introdued new generators and relations for Ariki-Koike algebras. This enable use to show the Frobenius formula for the characters of Ariki-koike algebras, which was only known for the case of type Au.II.Green function of type Au was introduced by Green in the study of representation theory of Ghn (Hg), and a combinatorial construction for it in terms of Schur-functions and Hall-Littlewood functions. Green fuctions for reductive groups in gewue was introduced by Deligne-Lusztig by using l-adis cohomology theory. In this research, we could extend the combinatorial approach as in the case of Au to the case of classical groups. Furthermore, our approach makes sense even is the case when weyl group is replaced by coplex reflection group G(r, 1, n).