Representation theory of algebraic groups via algebraic analysis

1. Tanisaki研究了量子化旗流形，特别是在1的根上。他制定了一个猜想，可以被视为一个类似的结果Bezrukavnikov，Mirkovic，Ryuminin关于对应的代表和D-模块的旗帜流形的积极特征。这与Backelin-Kremnitzer和Mirkovic最近的作品不同。虽然存在一些问题需要克服，但应该在不久的将来得到解决。他还延长了结果Soergel关于环的微分算子的部分旗流形的约化代数群，并获得了类似的结果微分算子作用于向量丛。此外，他还考虑了Soergel关于O.2范畴中心的结果的抛物型类似。Kashiwara证明了以基本权为其外权的仿射量子群的某些有限维表示的晶基与具有第1级最高权的不可约模的Demazure模的晶基同构。3. Ariki证明了经典Hecke代数的表示类型是由Poincare多项式决定的。Nakajima证明了Necrasov配分函数的第一项与Seidberg-Written的前势一致。Shoji解决了Lusztig关于有限域上特殊线性群特征标的猜想。此外，他还确定了出现在图中的标量。金田研究了正特征旗簇上的D-模与相应代数群的表示之间的对应关系。他制定了一定的衍生等价的算术微分算子由于贝特洛。并在射影空间的情形下得到了一些结果. Ichino研究了Saito-Kurokawa提升的对角限制，证明了某个L-函数的一个特殊值的代数性。

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.