Representation-theoretic study of spherical functions arising from number theory

Special functions (spherical functions) on algebraic groups play an important role in number theory, especially in the study of automorphic forms. In most cases, these spherical functions are related to spherical homogeneous spaces, such as symmetric spaces. In this research project, Kato (head investigator) studied spherical functions on spherical homogeneous spaces of reductive groups over non-archimedean local fields from a representation theoretic view point. The purpose of this research is two-fold : (1) To understand special functions such as zonal spherical functions or Whittaker functions in a uniform manner from the view point as above. (2) To obtain properties of these functions, including the uniqueness and explicit formulas, for important cases which arise in number theory. As for (1), we studied an orbit decomposition of spherical homogeneous spaces first. Then applying this, we obtained a general formula for spherical functions (at least in the case of symmetric spaces) t … More ogether with a method to compute the coefficients in this formula explicitly. As for (2), we got the uniqueness and an explicit formula for e.g. a symmetric space corresponding to quadratic base change by using the above mentioned method. This research is still under way. Other investigators obtained several results related to representation theory and spherical homogeneous spaces as follows. Saito studied zeta functions of prehomogeneous vector spaces, which is closely related to (spherical functions of) spherical homogeneous spaces, and showed the convergence and explicit formulas (in terms of local orbital zeta functions) in general. Matsuki investigated Weyl groups and Jordan decompositions arising from symmetric spaces. Nishiyama studied multiplicity free actions, which is a characteristic property of spherical homogeneous spaces, and the relation between theta correspondences and nilpotent orbits. Other investigators, Takasaki, Yamauchi et al. carried out researches on mathematical physics, automorphic forms and so on. Less

Special functions (spherical functions) on algebraic groups play an important role in number theory, especially in the study of automorphic forms. In most cases, these spherical functions are related to spherical homogeneous spaces, such as symmetric spaces. In this research project, Kato (head investigator) studied spherical functions on spherical homogeneous spaces of reductive groups over non-archimedean local fields from a representation theoretic view point. The purpose of this research is two-fold : (1) To understand special functions such as zonal spherical functions or Whittaker functions in a uniform manner from the view point as above. (2) To obtain properties of these functions, including the uniqueness and explicit formulas, for important cases which arise in number theory. As for (1), we studied an orbit decomposition of spherical homogeneous spaces first. Then applying this, we obtained a general formula for spherical functions (at least in the case of symmetric spaces) t … More ogether with a method to compute the coefficients in this formula explicitly. As for (2), we got the uniqueness and an explicit formula for e.g. a symmetric space corresponding to quadratic base change by using the above mentioned method. This research is still under way. Other investigators obtained several results related to representation theory and spherical homogeneous spaces as follows. Saito studied zeta functions of prehomogeneous vector spaces, which is closely related to (spherical functions of) spherical homogeneous spaces, and showed the convergence and explicit formulas (in terms of local orbital zeta functions) in general. Matsuki investigated Weyl groups and Jordan decompositions arising from symmetric spaces. Nishiyama studied multiplicity free actions, which is a characteristic property of spherical homogeneous spaces, and the relation between theta correspondences and nilpotent orbits. Other investigators, Takasaki, Yamauchi et al. carried out researches on mathematical physics, automorphic forms and so on. Less