Spherical functions for $GL_n$ over local fields, and summation of Hecke series

In §§ 1 and 2 explicit formulas for zonal spherical functions on the group $GL_n(D)$ are derived. Here $D$ is a division algebra of finite rank over a discrete normed field with finite residue class field. In § 3 these formulas are applied to summation of multiple Hecke series and zeta-functions in $n$ variables on the group $GL_n(D)$. In § 4 the results of § 3 are applied to summation of Hecke series and zeta-functions on the symplectic group of genus $n$ over local fields. Furthermore, the following conjectures are proved: the conjecture of Satake about the form of the denominator of zeta-functions, and the conjecture of Shimura about the degrees of the numerator and the denominator.
Bibliography: 7 titles.