Representation theory of finite algebraic groups

I.研究了复反射群G（r，1，n）及其相应的分圆Hecke代数（Ariki-Koike代数）。特别地，利用文献（Sakamoto-Shoji）证明的Schur-Weyl互易性，我们引入了Ariki-Koike代数的新生成元和新关系. II. Au型绿色函数是绿色在研究Ghn（Hg）的表示论时引入的，它是由Schur-函数和Hall-Littlewood函数组合构造的。Deligne-Lusztig利用l-adis上同调理论引入了格伍中还原群的绿色函数.在这项研究中，我们可以扩展的组合方法的情况下，Au的经典群。此外，我们的方法是有意义的，甚至是当Weyl群被替换为复反射群G（r，1，n）的情况下。

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).