Represontation theory of algebraic groups

1. Head Organizer和Masaki Kashiwara研究了仿射李代数上不可约最高权模的特征。首先证明了具有有理非临界最高权的不可约模的Kazhdan-Lusztig型特征标公式，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的不可约模的特征标，并研究其几何背景.广义超几何系统与Radon-Penrose变换：主要作者利用弱等变D-模理论研究了旗流形之间的Radon-Penrose变换，给出了在D-模理论意义下它是内射或满射的充分条件。此外，我还研究了每个个案中的条件。接下来的问题之一是确定Radon变换的图像和核。Head Organizer和Yoshiyuki Morita构造了抛物型预齐次向量空间坐标代数的量子形变。Morita进一步将这一结果应用于例外单李代数，并给出了例外预齐次向量空间的量子形变及其相对不变量的明确描述。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计数器部分，如zeta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.