Spherical functions for $GL_n$ over local fields, and summation of Hecke series
Sbornik. Mathematics, Tome 12 (1970) no. 3, pp. 429-452
Voir la notice de l'article provenant de la source Math-Net.Ru
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.
@article{SM_1970_12_3_a5,
author = {A. N. Andrianov},
title = {Spherical functions for $GL_n$ over local fields, and summation of {Hecke} series},
journal = {Sbornik. Mathematics},
pages = {429--452},
publisher = {mathdoc},
volume = {12},
number = {3},
year = {1970},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_1970_12_3_a5/}
}
A. N. Andrianov. Spherical functions for $GL_n$ over local fields, and summation of Hecke series. Sbornik. Mathematics, Tome 12 (1970) no. 3, pp. 429-452. http://geodesic.mathdoc.fr/item/SM_1970_12_3_a5/