Geodesic
Spherical functions for $GL_n$ over local fields, and summation of Hecke series
Sbornik. Mathematics, Tome 12 (1970) no. 3, pp. 429-452 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     author = {A. N. Andrianov},
     title = {Spherical functions for $GL_n$ over local fields, and summation of {Hecke} series},
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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/

[1] A. N. Andrianov, “Gipoteza Shimury dlya zigelevoi modulyarnoi gruppy roda $3$”, DAN SSSR, 177:4 (1967), 755–758 | MR | Zbl

[2] A. N. Andrianov, “Ratsionalnost kratnykh ryadov Gekke polnoi lineinoi gruppy i gipoteza Shimury o ryadakh Gekke simplekticheskoi gruppy”, DAN SSSR, 183:1 (1968), 9–11 | MR | Zbl

[3] A. N. Andrianov, “Teoremy ratsionalnosti dlya ryadov Gekke i dzeta-funktsii grupp $Gl_n$ i $Sp_n$ nad lokalnymi polyami”, Izv. AN SSSR, seriya matem., 33 (1969), 466–505 | MR | Zbl

[4] I. G. Macdonald, “Spherical functions on a $\mathfrak{p}$-adic Chevalley group”, Bull. Amer. Math. Soc., 74:3 (1968), 520–525 | DOI | MR | Zbl

[5] I. Satake, “Theory of spherical functions on reductive algebraic groups over $\mathfrak{p}$-adic fields”, Publ. Math. IHES., 18 (1963), 229–293 | MR | Zbl

[6] G. Shimura, “On modular correspondences for $Sp(n,\mathbf(Z))$ and their congruence relations”, Proc. Math. Acad. Sci. USA, 49:6 (1963), 824 | DOI | MR | Zbl

[7] T. Tamagawa, “On the $\zeta$-functions of a division algebra”, Ann. Math., 77:2 (1963), 387–405 | DOI | MR | Zbl