Representation theory of algebraic groups via algebraic analysis

1.谷崎研究了量子化旗形，特别是在1的根处。他提出了一个猜想，这个猜想可以看作是Bezrukavnikov-Mirkovic-Ryumin关于正特征的旗形上的表示与D-模的对应的一个类似的结果。这与巴克林-克雷姆尼策和米尔科维奇最近的作品不同。虽然还有一些问题需要克服，但它应该在不久的将来得到解决。他还推广了Soerel关于约化代数群的部分旗形上的微分算子环的一个结果，并对作用于向量丛的微分算子得到了类似的结果。此外，他还考虑了在范畴o2的中心抛物线类比Soerel的结果。Kashiwara证明了以基本权为外权的仿射量子群的有限维表示的晶基与1阶最高权的不可约模的Demazure模的晶基同构。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配分函数的第一阶与Seidberg-Written的PRO势重合。Shoji解决了Lusztig关于有限域上特殊线性群特征标的猜想。此外，他确定了猜想中出现的标量。Kaneda研究了具有正特征的旗簇上的D-模与相应代数群的表示之间的对应关系。由于Berthelot，他在算术微分算子方面建立了一定的导出等价，并在射影空间的情况下得到了一些结果。Ichino研究了Saito-Kurokawa提升的对角限制，并证明了一类L函数的特定值的代数性。

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.