Arithmetic of cohomological automorphic representations of orthogonal groups and theta series for indefinite quadratic forms

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We construct a kind of real analytic Siegel-Eisenstein series which associate to certain nontempered derived functor module of Sp (2n, R). A close study of its Fourier expansion gives us that the nontrivial Fourier coefficients are supported only for “indefinite" character of the Siegel unipotent subgroup. One can understand this fact to be related to a substancial invariant of the derived functor module, namely, the wave front set of its distribution character.We also give explicit formula of each Fourier coefficient, which enable us to compute its variously twisted Mellin transform, which give ar generalization of the works of Maass to a special real analytic automorphic representation. As a result we obtain a formula of it written by a Rankin-Selberg type Dirichlet series, where the coefficients include geodesic integrals of Maass wave forms. These study will be applied to construct a nontempered cohomological Saito-Kurokawa representation of Sp (2,A).