Represontation theory of algebraic groups

1. 仿射李代数上的最高权模首席组织者Masaki Kashiwara研究了仿射李代数上不可约的最高权模的性质。首先证明了具有有理非临界最高权的不可约模的Kazhdan-Lusztig型特征公式，然后将其推广到任意非临界权。接下来的问题之一是确定具有临界最高权值的不可约模的特征并研究其几何背景。广义超几何系统与Radon-Penrose变换。作者利用弱等变d模理论研究了旗流形之间的Radon-Penrose变换，并给出了在d模理论意义上它是内射或满射的充分条件。此外，我研究了每个个案的情况。接下来的问题之一是确定Radon变换的图像和核。主要组织者和Yoshiyuki Morita构造了抛物型预齐次向量空间坐标代数的量子变形。Morita进一步将这一结果应用于异常简单李代数，并给出了异常预齐次向量空间的量子变形及其相对不变量的显式描述。接下来的问题之一是研究与数论方向有关的预齐次向量空间理论的量子对抗部分，如ζ函数。

1. Highest weight modules over affine Lie algebrasThe head organizer and Masaki Kashiwara investigated on characters of irreducible highest weight modules over affine Lie algebras. We first proved Kazhdan-Lusztig type character formula for the irreducible modules with rational non-critical highest weights, and next generalized it to arbitrary non-critical weights. One of the next problems is to determine characters of the irreducible modules with critical highest weights and to investigate its geometyric back ground.2. Generalized hypergeometric systems and Radon-Penrose transforms.The head organizer investigated on Radon-Penrose transforms between flag manifolds using the theory of weakly equivariant D-modules, and gave sufficient conditions in order that it is injective or surjective in the sence of the D-module theory. Moreover, I studied the condition in each indivisual cases. One of the next problems is to determin the image and the kernel of the Radon transform.3. Quantum deformations of prehomogeneous vector spacesThe head organizer and Yoshiyuki Morita constructed quantum deformations for the coordinate algebras of the prehomogenous vector spaces of parabolic type. Morita further applied this result to exceptional simple Lie algebras, and gave an explicit description of the quantume deformation of the exceptional prehomogeneous vector spaces and its relative invariants.One of the next problems are to investigate of the quantum counter part of the theory of prehomogeneous vector spaces related to number theoretic direction such as zeta functions.