On zeta functions of prehomogeneous vector spaces

(1)Saito给出了约化不可约正则准齐次向量空间的Zeta函数的一个完全一般性的证明，证明中所用的方法似乎适用于更一般的情况，例如正则准齐次向量空间.(2)Saito显式地确定了非阿基米德局部域上与对称矩阵(Sato-Kimura分类中的(15)型)表示有关的齐次向量空间的函数方程，这一结果在非阿基米德局部域上的代数群的表示上有一定的应用。(3)Kato证明了环面的一个子半群可以被选择作为p-进约化群关于其球面群和极大紧群的双陪集分解的代表。这一结果似乎在非阿基米德局部域上准齐次向量空间的Zeta函数的计算中有一定的应用。(4)Matsuki证明了紧李群关于两个对合的双陪集分解的分解定理、分类和根系理论，并计算了余维为1的旗形的轨道分解的几个例子。(5)Nishiyama得到了半单群的Bernstein度的积分表示。这使得在某些特殊情况下计算伴随圈成为可能。(6)Nishiyama计算了经典Gorup情形下Weyl群表示的Kawanaka不变量。(7)Yoshino成功地发展了超曲面奇点局部环上的Cohen-Macaulay模的分类理论，并给出了利用Cohen-Macualay逼近对一般模进行分类的方法。

(1) Saito gave a proof for the convergence of zeta functions of reduced irreducible regular prehomogeneous vector spaces in full generality, The method used in this proof seems to be applicable to more general cases, for example, to the case of regular prehomogeneous vector spaces.(2) Saito determined the functional equation explicitly in the case of homogeneous vector spaces over non-archimedean local fields which are related to the representation of symmetric matrices(type (15) in the classification of Sato-Kimura), This result has some applications to the representation of algebraic gorups over non-archimedean local fields.(3) Kato showed that a subsemigroup of a torus can be chosen as representatives in the doule coset decomposition of a p-adic reductive group with respect to its spherical group and its maximal compact group. This result seems to have some applications in the computation of zeta functions of prehomogeneous vector spaces over non-archimedean local fileds.(4) Matsuki showed a decomposition theorem, a classification and a theory of root systems in the double coset decomposition of compact Lie groups with respect to two involutions, and calculated several examples of orbit decomposition of flag manifolds in the case of codimension one.(5) Nishiyama obtained an integral representation for the Bernstein degrees for semi-simple groups. This made it possible to calculate the associated cycles in some special cases.(6) Nishiyama computed the Kawanaka invariant of the representation of Weyl groups in the case of classical gorups.(7) Yoshino succeeded in developing the theory of classification of Cohen-Macaulay modules over the local rings of singular points of hypersurfaces by means of linkage, and gave a method to classify general modules by using the Cohen-Macualay approximation.