Representation theory of algebraic groups through algebraic analysis

1. Kac-Moody李代数上的最高权模。kashiwara和Tanisaki试图确定具有临界最高权的仿射李代数上不可约模的性质。特别地，他们考虑了带d模的仿射李代数在半无穷标志流形上的表示关系。通过调查，他们注意到等变线束的行为与普通旗形上的行为有很大的不同。量子群的标志流形stanisaki构造了对应于一般抛物子群的广义标志流形。治疗无能的根治性肿瘤要比治疗普通病例更加细致。此外，Morita和Tanisaki还试图从非交换方案的角度寻找量化标志流形上的束上同调和d模的良好表述。利用玻色子构造了环面李代数的若干表示。他还研究了环面李代数的自同构，并发现了环面李代数与模群的联系。量子群在Laurent多项式环上的表示。skaneda研究了量子群在Laurent多项式环上的表示，并证明了kempv定理的一个版本。kawanaka对Sato的游戏进行了推广，并从表征理论的角度对其进行了研究。shoji试图将A型Hecke代数的Frobenius公式推广到复反射群的Hecke代数中，并在Ariki-Koike代数的情况下取得了成功。

1. Highest weight modules over Kac-Moody Lie algebrasKashiwara and Tanisaki tried to determine characters of irreducible modules over affine Lie algebras with critical highest weights. In particular, they considered relations of representations of affine Lie algebras with D-modules on the semi-infinite flag manifold. Through the investigation they noticed that the behavior of the equivariant line bundles is quite different from those on the ordinary flag manifold.2. Flag manifold for quantum groupsTanisaki constructed generalized flag manifold corresponding to general parabolic subgroups. The treatment of the unipotent radical is more delicate than the ordinary case. Besides Morita and Tanisaki tried to find a good formulation for the sheaf cohomologies and D-modules on the quantized flag manifold from the view point of non-commutative schemes.3. Representations of toroidal Lie algebrasSaito constructed certain representations of toroidal Lie algebras usig Boson. He also investigated the automorphisms of the toroidal Lie algebras and found a connection with the moudular groups.4. Representations of quantum groups over Laurent polynimial ringsKaneda investigated representations of quantum groups over the Laurent polynimial rings, and have proved a version of a theorem of Kempf.5. Solvable gamesKawanaka found a generalization of Sato's game, and investigated on it from the view point of representation theory.6..Representations of comples reflection groups and their Hecke algebrasShoji tried to extend the Frobenius formua for the Hecke algebra of type A to the Hecke algebras for complex reflection groups, and succeed in it in the case of Ariki-Koike algebra.