Spherical functions of Hermitian and symmetric forms. III (Q1121478)

[For part I see the preceding review Zbl 0674.43006.] This paper is the third part of a series of papers, whose aim is to show the functional equation for spherical functions and determine their possible poles for the space of nondegenerate Hermitian, or symmetric matrices over a p-adic number field. With the notation of part I the following theorem is proved. (1) Let \({\mathcal S}_ n\) be the symmetric group on n letters. For each \(\sigma\in {\mathcal S}_ n\), there exists a matrix C(\(\sigma\),z) in \(GL_{4^ n}({\mathbb{C}}(q^{z_ 1}...,q^{z_ n}))\) such that \[ L(x;\chi;\sigma (z))=\sum_{\chi}C(\sigma,z)_{\chi,\chi '}L(x,\chi ',z). \] (2) The function \(\prod_{1\leq i,j\leq n,i\neq j}(q^{2z_ i}-q^{2z_ j- 1})L(x;\chi;z)\) is a polynomial in \(q^{\pm z_ 1},...,q^{\pm z_ n}.\)