Geometric Representation theory of reductive group

还原李群的幺正最高权表示的Bernstein度、相关环和各向同性表示是一类重要的几何不变量。对于辛群，Ochiai和K. Nishiyama确定了Bernstein度。对于正交群和酉群，Nishiyama， Ochiai和K. Taniguchi确定了Bernstein度和相关环。一般情况下，包括特殊群体是由Ochiai和Shohei Kato完成的。Ochiai和Kato将这一结果推广到描述一组无多重表示的球轨道度。这是selberg型积分的一个新应用。表示被认为是函数空间的量化。为了同时处理普适包络代数和坐标环，Oshima引入了同质包络代数的概念。使用Capelli算子，这是一个小量的量子化，Oshima描述了标量型广义Verma模的湮灭子理想。与这项工作相关，Ochiai描述了包络代数和不变微分算子环的中心的区别。在此基础上，Oshima和Shimeno对非极小抛物线的泊松变换图像进行了表征。Wakimoto讨论了各种仿射代数的模不变性。Kaneko从有限域上奇异模和超奇异椭圆曲线的几何角度研究了模函数*j。Ochiai还讨论了具有特定类型分支指数的椭圆曲线的生成函数的准模性。Konno考虑了局部域上约化群的单幂表示，特别是给出了低阶群的单幂表示的一个特殊子类CAP表示的描述。

The Bernstein degree, associated cycles and isotropy representation for a unitary highest weight representation of reductive Lie groups are important family of geometric invariants. For the symplectic groups, Ochiai with K. Nishiyama determines the Bernstein degree. For orthogonal and unitary groups, Nishiyama, Ochiai and K. Taniguchi determines the Bernstein degree and associated cycles. The general case including exceptional groups are done by Ochiai and Shohei Kato. This results are generalized to describe the degree of spherical orbits of a family of multiplicity-free representations by Ochiai and Kato. This is a new application of the Selberg-type integral.A representation is considered as a quantization of a function space. Oshima introduces the notion of homogenized enveloping algebra, in order to deal with universal enveloping algebras and coordinate rings simultaneously. Using the Capelli operator, which is a quantization of minors, Oshima describes the annihilator ideals of generalized Verma modules of scalar type. Related to this work, Ochiai describes the difference of the centers of the enveloping algebras and the rings of invariant differential operators. Based on the work above, Oshima with Shimeno characterize the image of the Poisson transform associated to non-minimal parabolic.Wakimoto discuss the modular invariance of characters of various affine algebras. Kaneko investigates the modular function *j from the geometry of singular moduli and supersingular elliptic curves over finite fields. Ochiai also discuss the quasi-modularity of the generating functions of an elliptic curve with a specified type of ramification indeces. Konno considers the unipotent representations of reductive groups over local fields, especially gives the description of CAP representations, which is a special subclass of unipotent representation, for low-rank groups.