Representation theory of algebraic groups via algebraic analysis

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1. Tanisaki investigated on the quantized flag manifolds, especially at roots of 1. He has formulated a conjecture which can be regarded as an analogue of the result of Bezrukavnikov-Mirkovic-Ryuminin about the correspondence of representations and D-modules on the flag manifold in positive characteristics. This is different from recent works of Backelin-Kremnitzer and Mirkovic. It should be solved in the near future although there are some problems to be overcome. He also extended a result of Soergel about the ring of differential operators on the partial flag manifold of reductive algebraic groups and obtained similar results for differential operators acting on vector bundles. Furthermore, he considered about parabolic analogue of Soergel's result on the center of category O.2. Kashiwara showed that the crystal base of some finite dimensional representation of affine quantum group with fundamental weight as its external weight is isomorphic to that of the Demazure module of irreducible module with level 1 highest weight.3. Ariki has shown that the representation types of the classical Hecke algebras are governed by the Poincare polynomials.4. Nakajima proved that the first tern of the Necrasov partition function coincides with the pro-potential of Seidberg-Written.5. Shoji has solved Lusztig's conjecture on the characters of the special linear groups over finte fields. Moreover, he determined the scalar appearing in the conjecture.6. Kaneda investigated on the correspondence between D-modules on flag varieties in positive characteristics and representations of the corresponding algebraic groups. He formulated a certain derived equivalence in terms of arithmetic differential operators due to Berthelot., and obtained some results in the case of the projective space.7. Ichino investigated on the diagonal restriction of Saito-Kurokawa lift, and proved the algebraicity of a special value of a certain L-function.