Zeta functions on weakly spherical homogeneous spaces

1.Sato, the head investigatorstudied the weakly spherical homogeneous spaces (WSHS for short) obtained from the prehomogeneousvector spaces (PV for short) (Spin D210 - D2 × D23 - D2, half-spin【cross product】Λ - D21 - D2)，(SL - D25 - D2 × GL - D23 - D2Λ - D22 - D2【cross product】Λ - D21 - D2) and proved the functional equations satisfied by Eisensteinseries attached to the spaceswhich is a joint work with T. Kimura of Tsukuba Univ. and his students. As an application we cancalculate explicitly the Fourier transforms of the complex powers of relative invariants of the PV'sThis shows that the theory of WSHS is very fruitful . This shows that the theory of WSHS is very fruitful因为这两个PV的结构是非常复杂的，甚至是micro local的方法calculus can not be applied to them. We also made several attempts to generalize the theory to WSHSreductive groups other than the general linear group. the result is still unsatisfactory；however several suggestive partial results have been obtained.2.Hiron…aka succeeded in calculating the explicit formula for spherical functions of sphericalhomogeneous space over p-adic fields in a rather general setting. Using the formula，she constructed the theory of spherical functions of the space of hermitian forms over unramifiedquadratic extensions of the base p-adic field and gave an explicit formula for local densities ofhermitian forms. Moreover, jointly with Sato，she calculated local densities of quadratic forms over nondyadic p-adic fields explicitly in themost general settingresults can be extended to the space of hermitian forms over ramified quadraticArakawa studied mainly Koecher-Maass zeta functions attached to Siegel modular formsand obtained some results包括following：a) a generalization of Koecher-Maass zeta functions to Jacobi formsb) an Koecher-Maass zeta function attached to the Nagaoka Eisenstein explicit expressionseries of weight 1 of degree 2，其中包含了来自p-adic Siegel Eisenstein series. Less

1.Sato, the head investigator, studied the weakly spherical homogeneous spaces (WSHS for short) obtained from the prehomogeneous vector spaces (PV for short) (SpinィイD210ィエD2 × GLィイD23ィエD2, half-spin 【cross product】 ΛィイD21ィエD2), (SLィイD25ィエD2 × GLィイD23ィエD2, ΛィイD22ィエD2 【cross product】 ΛィイD21ィエD2) and proved the functional equations satisfied by Eisenstein series attached to the spaces, which is a joint work with T. Kimura of Tsukuba Univ. and his students. As an application we can calculate explicitly the Fourier transforms of the complex powers of relative invariants of the PV's above. This shows that the theory of WSHS is very fruitful, since the structure of these two PV's is very complicated and even the method of micro local calculus can not be applied to them. We also made several attempts to generalize the theory to WSHS of reductive groups other than the general linear group. The result is still unsatisfactory ; however several suggestive partial results have been obtained.2.Hiron … More aka succeeded in calculating the explicit formula for spherical functions of spherical homogeneous spaces over p-adic fields in a rather general setting. Using the formula, she constructed the theory of spherical functions of the space of hermitian forms over unramified quadratic extensions of the base p-adic field and gave an explicit formula for local densities of hermitian forms. Moreover, jointly with Sato, she calculated local densities of quadratic forms over nondyadic p-adic fields explicitly in the most general setting ; the results can be extended to the space of hermitian forms over ramified quadratic extensions.3.Arakawa studied mainly Koecher-Maass zeta functions attached to Siegel modular forms and obtained some results including the following : (a) a generalization of Koecher-Maass zeta functions to Jacobi forms, (b) an explicit expression of the Koecher-Maass zeta function attached to the Nagaoka Eisenstein series of weight 1 of degree 2, which is obtained from p-adic Siegel Eisenstein series. Less