Representation Theoretic and/or Geometric Research for Theta Series

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(1) The classical correspondence between Jacobi forms and Sigel cusp forms of half-integral weights is studied from representation theoretic point of view. The basic tool is Well representation. The results are published on "On Siegel modular forms of half-integral weights and Jacobi forms" (Trans. A.M.S.351 (1999), pp.735-780).(2) Hermite polynomials of multi-variables are defined in two ways through a detailed study of the irreducible decomposition of the Weil representation of Sp(n, *) restricted to the dual pair (U(n), U(1)). As K-type vectors for K = U(n), we will get products of the classical (one-variable) Hermite polynomials which give a complete system of the solutions of the Schrodinger equation of n-dimennsional harmonic ascillator. On the other hand, as K-type vectors for K = U(1), we will get another complete system of the solution of the Schrodinger equation which is not of separated variables, The results will be published on the paper "K-type vectors of Weil representat … More ion and generalized Hermite polynomials".(3)Weil's generalized Poisson summation formula, which is valid only for theta group, is extended to the general paramodular groups. As applications ; 1) a representation theoretic proof of the transformation formula of Riemann's theta series, and 2) the transformation formula of theta series associated with a integral quadratic form with harmonic polynomials. The results will be published on the paper "On an extension of generalized Poisson summation formuls of Weil and its applications".(4) We applied the method of T.Shintani (J.Fac. Sci. Univ. Tokyo 22 (1975), pp. 25-56) to the general semi-simple algebraic group over *, and found that a part of the dimmension formula of the space of the automorphic forms attached to an integrable representaton is given by a special values of the zeta functions of pre-homogeneous vector space of parabolic type srising from a maximal parabolic subgroup defined over *. Also we found that there seems to exist an interesting relationship between the non-zero set of the Fourier tranform of the spherical trace function of the integrable representaiton and the Zariski open orbit of the pre-homogeneous vector space. A part of the results will be published on the proceeding of the Autumn Workshop on Number Theory at Haluba (1998). Less