On zeta functions of prehomogeneous vector spaces

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(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.