Representation theory of algebraic groups through algebraic analysis

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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.