SPHERICAL FUNCTIONS FOR<i>GL_n</i>OVER LOCAL FIELDS, AND SUMMATION OF HECKE SERIES

Abstract

In ?? 1 and 2 explicit formulas for zonal spherical functions on the group GLn(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 GLn(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 items.