Geometric Representation theory of reductive group

约化李群的么正最高权表示的Bernstein度、伴随圈和迷向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度和相关圈。包括例外组在内的一般情况是由Ochiai和Shohei Kato完成的。这一结果被推广到描述Ochiai和Kato的一族无重数表示的球面轨道的次数。这是Selberg型积分的一个新的应用，其表示被认为是函数空间的量子化。为了同时处理泛包络代数和坐标环，Oshima引入了齐次化包络代数的概念。利用子式的量子化形式Capelli算子，Oshima刻画了标量型广义Verma模的零化子理想。与此相关的是，Ochiai刻画了包络代数的中心与不变微分算子环的中心之差。在上述工作的基础上，Oshima和Shimeno刻画了与非极小抛物有关的泊松变换的像，并讨论了各种仿射代数特征标的模不变性。Kaneko从有限域上奇异模和超奇异椭圆曲线的几何出发，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母函数的拟模性质。Konno考虑了局部域上约化群的幂等表示，特别是给出了低秩群的一类特殊的幂表示--CAP表示的刻画。

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.